1404notes.pdf

MAS1404/EEE2011. Discrete mathematics for computing science

Sarah Rees

[email protected]

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Recommended books

J.GerstingMathematical structures for computer science, 4th edition(Freeman, 1999).

R.P.GrimaldiDiscrete and combinatorial mathematics, 4th edition (Addison-Wesley, 1999).

J.K.TrussDiscrete mathematics for computer scientists, 2nd edition(Addison-Wesley, 1999).

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Syllabus

Set Theory: defining sets by listing, by properties, and by induction;unions, intersections, complements; Venn diagrams; power sets. Carte-sian products.

Relations: definition and examples of unary, binary and tertiary relations.Representation of binary relations using directed graphs. Building moreuseful structures: cartesian products; lists; inductive definition of struc-tures.

Functions: domain, codomain, range of a function; injections, surjections,bijections; inverse functions; total and partial functions; recursive func-tions.

Graphs: directed, undirected. Basic properties. Trees, parsing trees.

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Reasoning about structures using structural induction;

Propositional Calculus: propositional connectives, formulas, truth tables,tautology.

Predicate Logic: existential and universal quantifiers; predicate calculus.

Boolean Algebra: axioms; examples; simplification of Boolean expressions.

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Schedule of lectures and coursework

Lecture 1 Sets 1.1Lecture 2 1.2,1.3Lecture 3 1.41.6Lecture 4 Relations 2.12.3Lecture 5 2.4,2.5Lecture 6 Functions 3.13.3Lecture 7 3.4, 3.5Lecture 8 3.5ctd,3.6Lecture 9 3.73.9

Lecture 10 Graphs 4.1, 4.2Lecture 11 4.34.6Lecture 12 Recursion&induction 5.1,5.2Lecture 13 5.2ctd.Lecture 14 Prop. Logic 6.16.4Lecture 15 6.56.8Lecture 16 Predicate Logic 7.17.4Lecture 17 Boolean Algebra 8.18.5Lecture 18 Boolean Algebra 8.68.9

The remaining hours will be used for recap sessions.

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Homework 1 Sets and Relations Chapters 1,2Homework 2 Functions Chapter 3Homework 3 Graphs, recursion and induction Chapters 4,5Homework 4 Logic Chapters 6,7,8

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1 Sets 14

1.1 Definition and description of a set . . . . . . . . 14

1.2 Basic set operations and relations . . . . . . . . 23

1.3 Rules relating the set operations . . . . . . . . . 30

1.4 Venn diagrams . . . . . . . . . . . . . . . . . . . . 33

1.5 Power sets . . . . . . . . . . . . . . . . . . . . . . . 42

1.6 Direct products . . . . . . . . . . . . . . . . . . . . 43

2 Relations 46

2.1 Definition and examples . . . . . . . . . . . . . . 46

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2.2 Binary relations; infix notation . . . . . . . . . . 51

2.3 Diagram of a binary relation . . . . . . . . . . . 52

2.4 Properties of a binary relation . . . . . . . . . . 53

2.5 Equivalence relations, partial orders and theirdiagrams . . . . . . . . . . . . . . . . . . . . . . . . 56

3 Functions 61

3.1 Definition of a function, notation, examples . . 61

3.2 The graphs of a function, directed and Cartesian 67

3.3 Functions vs. non-functions . . . . . . . . . . . . 72

3.4 Composing functions . . . . . . . . . . . . . . . . 74

3.5 Inverse functions . . . . . . . . . . . . . . . . . . . 79

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3.6 Properties that make a function invertible . . 85

3.7 Computing an inverse for a function . . . . . . 91

3.8 Partial functions . . . . . . . . . . . . . . . . . . . 92

3.9 Functions defined recursively . . . . . . . . . . . 97

4 Graphs 102

4.1 Definition and examples, notation . . . . . . . . 102

4.2 Paths, connectivity, components . . . . . . . . . 110

4.3 Bipartite graphs . . . . . . . . . . . . . . . . . . . 113

4.4 Trees and forests . . . . . . . . . . . . . . . . . . . 118

4.5 Parsing trees . . . . . . . . . . . . . . . . . . . . . 122

4.6 Spanning trees . . . . . . . . . . . . . . . . . . . . 1259

5 Recursion and induction 127

5.1 Examples of some sets defined inductively . . . 127

5.2 Some examples of proof by induction . . . . . . 130

6 Propositional logic 139

6.1 Statements, propositions, paradoxes . . . . . . 139

6.2 Compound propositions, logical connectives . 142

6.3 Logical equivalence, truth tables . . . . . . . . . 145

6.4 Construction of truth tables . . . . . . . . . . . . 147

6.5 Rules relating the operations of propositionallogic . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

6.6 Tautologies and contradictions . . . . . . . . . . 156

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6.7 Translating English sentences into the languageof propositional logic . . . . . . . . . . . . . . . . 157

6.8 The connectives , and . . . . . . . . . . . 159

7 Predicate logic 162

7.1 The universal quantifier . . . . . . . . . . . . . . 166

7.2 The existential quantifier . . . . . . . . . . . . . . 169

7.3 Multiple quantifiers . . . . . . . . . . . . . . . . . 171

7.4 Relating the existential and universal quantifiers172

7.5 Translating English sentences using quantifiers 173

7.6 A systematic approach to translation . . . . . . 175

7.7 First we find expressions for the components . 176

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7.8 Quantifying more than one variable . . . . . . . 177

7.9 Translating whole statements . . . . . . . . . . . 178

8 Boolean algebra 185

8.1 Definition of a Boolean algebra . . . . . . . . . . 185

8.2 Examples of Boolean algebras . . . . . . . . . . 187

8.3 Principle of duality for Boolean algebras . . . . 189

8.4 Rules deducible from the axioms . . . . . . . . . 190

8.5 Verifying the extra rules . . . . . . . . . . . . . . 191

8.6 Boolean expressions . . . . . . . . . . . . . . . . . 194

8.7 Tidying up Boolean expressions . . . . . . . . . 196

8.8 Disjunctive normal form . . . . . . . . . . . . . . 19812

8.9 Relating disjunctive normal form and truth ta-bles . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

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1 Sets

1.1 Definition and description of a set

A set is a collection of objects (known as the elements) for which mem-bership is decidable.

The definition is a bit mysterious, e.g. what do we mean by the phrasefor which membership is decidable?

Basically a set is a collection of things. It might be described explicitly,by listing its elements, but it might be described differently.

Well start by looking at some examples, described in various differentways.

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Examples 1.1 (Different types of sets)

Sets described by listing their elements , between brackets {, }and separated by commas, e.g. {1, 3, 5, 7, 9}, the odd digits.We read { as the set (consisting) of, and keep quiet as we read }, soread the above as the set consisting of 1, 3, 5, 7, 9.

Each element is counted once only (if it were convenient, we mightwrite an element down more than once) and the order the elements arelisted in is irrelevant.

Two sets are equal if they contain the same elements. So

{1, 3, 5} = {3, 1, 5} = {1, 1, 3, 5}.We read as is an element of, 6 as is not an element of

1 {1, 3, 5}, 3 {1, 3, 5}, 4 6 {1, 3, 5}15

Other examples of sets:

{cat, horse, dog }, {0, 1, cat,monkey, 57}

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Some standard sets :Z, the integers, Q, the rationals (all fractions),R, the reals (all numbers met in nature, positive or negative, rationalor irrational),C, the complex numbers, N, the natural numbers,, the empty set.NB: In this course, we shall define 0 to be a natural number, so

N = {0, 1, 2, 3, . . .}.But notice that some books exclude 0 from N, call 1 the first naturalnumber. We call the set {1, 2, 3, . . .}, formed from N by excluding 0,the positive integers, or Z+.

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Sets described by rules , such as {x Z : x > 10}, which we readas the set of x in the integers such that x is greater than 10. We read: as such that, so that, for which, as is convenient. The symbol |may be used instead of :; it means exactly the same. where : or | assuch that.

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Sets described inductively/recursively :The set of all integer powers of 2, {2n : n N} can also be defined asthe smallest set S such that S contains 1, and if x S then 2x S.Similarly,

N can be defined by the statement:N contains 0, and if x N then x+1 N, but nothing else is in N..Z+ can be defined by the statement:Z+ contains 1, and if x Z+ then x+ 1 Z+, but nothing else is inZ+.

And the set S defined by the rule:S contains 10, and if x S then x+ 1 S, but nothing else is in Sis simply {x Z : x 10}.The fact that these sets can be defined in this way means it is easyto construct them, deduce and prove properties using induction (well

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meet this later), and to define functions over them.

Similarly, inductive definitions of trees and linked list structures makeit easy to work with those.

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A collection of elements that is not properly defined is nota set!

For a collection of elements to be a set, membership must be decidable.Consider

, the collection of all sets A such that A 6 A.Is an element of ? We try to decide. Lets see what happens:-

if is an element of , then . . .

if is not an element of , then . . .

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Both possibilities lead us to nonsense. So in fact, cant be a set; the defi-nition is inconsistent. This strange situation is known as Russells paradox.

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1.2 Basic set operations and relations

Definition 1.2 (Cardinality)We define the size (cardinality) of a setto be the number of elements in it, and use the Notation |A|.A set might have infinite cardinality. Note too that even if an element islisted more than once in the description of a set it is only counted once.

Definition 1.3 (subspace) Suppose that A,B are sets. We say thatA is a subset of B and write A B or A B if every element of A isalso in B.

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Notice that A A and A.We can write the symbols and backwards as and . Then B Aand B A mean exactly the same as A B and A B. If A is not asubspace of B, then we write A 6 B.

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Definition 1.4 (Binary operations)We define the intersection of Aand B, written A B, to be the set of all elements of A that are also inB.

We define the union of A and B, written A B, to be the set whoseelements consist of all the elements of A together with all the elementsof B.

We define the set difference A \B to be the subset of A formed from Aby deleting all elements also in B.

We define the symmetric difference A+B to be the union of A \B andB \ A.Instead of + as a notation for symmetric difference, we also see 4 and.

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Definition 1.5 (Universal set, complement)We define a universalset U to be the set that contains as an element everything we are currentlyinterested in. (e.g it might be the set of all people on a database). Thenwe define the complement Ac of a set A to be the set U \ A.We can illustrate the definitions of these operations with some examples:

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Notice that some different combinations of operations always give thesame answer e.g.

A (B C) = (A B) (A C)as we can verify by working an example. Of course this isnt just a coin-cidence . . .

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1.3 Rules relating the set operations

Commutative laws: A B = B A A B = B AAssociative laws: A (B C) = A (B C) =

= (A B) C = (A B) CDistributive laws: A (B C) = A (B C) =

(A B) (A C) (A B) (A C)U and : A U = A A = AComplements: A Ac = A Ac = UDe Morgans laws: (A B)c = Ac Bc (A B)c = Ac BcIdempotent laws: A A = A A A = A

(Ac)c = A

We can deduce further rules as simple consequences of these, such as

(A B C)c = Ac Bc Cc,30

which we get by applying the first of De Morgans laws twice: As anexample, we verify this last rule for a particular choice of U , A,B,C.

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Note that rules relating set operations generally come in pairs. Given onerule we get a second usually distinct rule by swapping with and with U . The principal of duality makes this work. That says that for anystatement that is true about sets in general, the statement that you getfrom that by replacing each by a , each by a , each U by a andeach by a U is also true. The second statement is called the dual of thefirst. If the second statement is equal to the first, then the statement issaid to be self-dual; e.g. (Ac)c = A.

Example:. The dual of

(A B C)c = Ac Bc Ccis

(A B C)c = Ac Bc Cc

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1.4 Venn diagrams

Venn diagrams provide a useful technique to verify rules that hold in settheory. A Venn diagram for a collection of sets A,B,C, . . . is a diagramthat displays each set as a enclosed region within the plane. Usuallywe draw a rectangular region representing the universal set U and thenrepresent each of the sets A,B,C, . . . as connected regions within therectangle. An intersection of two of those sets, such as AB, is displayedas the region common to both the regions representing the two sets (Aand B).

As examples, well draw the Venn diagrams

(a) for a single set A within the universal set. In that diagram we caneasily identify A and Ac.

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(b) two subsets A,B of the universal set. In that diagram we can easilyidentify A B, A B, A \B, B \ A, A +B Ac, etc.

(c) three subsets A,B,C of the universal subset.

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(d) four subsets A,B,C,D of the universal subset. By now its startingto get a bit trickier . . .

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We can use a Venn diagram to verify a general law about sets by drawingthe sets A,B,C, . . . in the most general configuration possible (ie so thateach possible intersection between the sets is represented by a distinct non-empty region), and then using shading to help us construct and comparethe various sets that are involved in the law.

Example: we verify the law

(A B)c = Ac Bcby drawing two Venn diagrams, as below

In the first we identify the set (A B)c. We shade A B, and then seeits complement as the unshaded portion of the diagram.

In the second we shade Ac one way, Bc another way, and then identifytheir union as the total shaded area. The fact that the regions identifiedin the two diagrams match up verifies that the two sets are equal.

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Example: we verify the law

(A B) C = (A C) (B C)Again we use 2 diagrams, as below.

In the first, we mark AB and C, by shading in two different ways, andthen identify (A B) C as the total shaded area.In the second we mark A C by shading one way, B C by shadinganother way, and then their intersection as the area that is shaded bothways. Since we can see that the regions identified in the two diagramsmatch up, we have verified that the two sets are equal.

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We can also use Venn diagrams to deduce some properties of a particularconfiguration of sets from others.

For example, suppose that we have three sets A,B,C for which B Aand A C = . Then we can deduce that B C = .We draw a Venn diagram for A,B,C that incorporates the informationthat B A by putting the region representing B within the regionrepresenting A. Then since AC = we draw the region representing Cas disjoint from the region representing A. Then it is necessarily disjointfrom the region representing B,and hence B C = .But notice that if now D is a fourth set that contains A, DC need notbe empty!

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1.5 Power sets

The power set of A, known as P(A), or (sometimes) 2A. is the set of allsubsets of A.

Where A has n elements, its power set has 2n.

As examples we write down P({1, 2}) and P({1, 2, 3}) and count theirelements.

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1.6 Direct products

The direct product of two sets A,B is the set

{(a, b) : a A, b B}.More generally, the direct product of n sets A1, A2, . . . An is the set

{(a1, a2, . . . an) : a1 A1, a2 A2, . . . an An}.Order of composition is important, i.e. in general AB is not the sameas B A.Some examples:-

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Direct products are also called Cartesian products.

The sets A1, . . . An are called the components of the cartesian productA1 . . . An.The size of A1 A2 . . . An is equal to the product of the size of itscomponents, ie

|A1 A2 . . . An| = |A1| |A2| |An|.We often store data in sets that are, in effect, direct products, althoughwe may not call them such. We might call such sets records, or structures.We usually access the components of a record using dot notation.

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2 Relations

2.1 Definition and examples

Basically a relation is a rule or property that holds for some elements ofa set but not for others, or between some (ordered) pairs of elements,or triples of elements but not for others. In order to talk sensibly aboutrelations we need a proper, more formal definition.

Definition 2.1 (relation) Given sets A1, A2, . . . An, we define an n-ary relation R between A1, A2, . . . An to be a subset of A1A2 . . . An.If we have just one set, that is, if n = 1, then R is a unary relation.

If n = 2, then we have a binary relation.

If n = 3, then we have a ternary relation.

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Often A1, A2, . . . An are all equal to the same set A, in which case we cansay our relation is on A. Where R is an n-ary relation and (x1, . . . xn) R, we say that R holds at (x1, . . . xn). Sometimes we write R(x1, . . . xn)rather than (x1, . . . xn) R.In this course most of the relations we shall meet will be either binary orunary. A few will be ternary.

We also use the terms property and predicate to mean exactly the samething as relation. In particular we usually use the term property for aunary relation, and well meet the term predicate later when we do somelogic.

Some unary relations :On Z, x is even, x is positive. We can write the statement x is evenin set theory language, as x EvenInt or in the form EvenInt(x).

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Similarly we can write the statement x is positive as x PosInt or asPosInt(x).

For the set A = {1, 2, 3, 4, 5}, the relation EvenInt on A is the subset{2, 4}.

Some binary relations :On Z, we have x + y is even, x is greater than y. We can write thestatement x+y is even in set theory language as (x, y) EvenSumor EvenSum(x, y). For the set A = {1, 2, 3, 4, 5, }, the relationEvenSum on A is the set of pairs

For the same set the relation IsGreaterThan is the set of pairs

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Where is any set, for x , A x is an element of A,defines a binary relation between and P(). For = {1, 2, 3}, wehave

P() =and hence the relation is the set of pairs

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A ternary relation :On Z, SumEquals(x, y, z) to mean x + y = z. For the set A ={1, 2, 3, 4, 5, }, the relation SumEquals on A is the set of triples

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2.2 Binary relations; infix notation

Binary relations are particularly useful, and since a binary relation relatestwo variables, it is often pleasant to use a different kind of notation todenote them, which we call infix notation. In this notation, given a binaryrelation R, rather than write (a, b) R, or R(a, b) (prefix notation), wewrite instead aRb.

There are some very common binary relations, many of which have specialnon-alphabet characters as their most natural labels.

= .

2.3 Diagram of a binary relation

We often use diagrams to illustrate binary relations, points with directedarcs joining them. (These are special cases of directed graphs, which wellmeet later.)

Given a binary relation R on a set A, we draw a node for each elementof A, and then draw a directed arc from x A to y A if xRy.As an example, well draw the diagram of on {1, 2, 3, 4, 5}

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2.4 Properties of a binary relation

A relation R on A is reflexive if for all x A, xRx. On the diagram forR, this means that every node is connected directly to itself.

R is symmetric if for all x, y A with xRy, we also have yRx. On thediagram for R, this means that whenever there is a directed edge from xto y there is also one from y to x.

R is transitive if for all x, y, z A with xRy and yRz we also have xRz.On the diagram for R this means that whenever there is a directed pathfrom x to z via y there is also a directed edge from x to z.

R is antisymmetric if whenever both xRy and yRx are true we havex = y. On the diagram for R this means that we have at most onedirected edge between 2 distinct vertices.

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2.5 Equivalence relations, partial orders and their diagrams

We call a relation that is reflexive, symmetric and transitive an equivalencerelation. As an example we have m, for each ms, on Z or any other setof integers.

We call a relation that is reflexive, antisymmetric and transitive a partial order.As examples, we have on Z, on the set of subsets of a set .For equivalence relations and partial orders we have special, more compactdiagrams.

For an equivalence relation we form the equivalence graph by deleting allthe loops from the standard diagram of the relation, and replacing eachpair of directed edges between two related vertices by a single undirectededge.

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As an example we can draw both the standard diagram and the equivalencegraph of 3 on {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.

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For a partial order we form the Hasse diagram by deleting all the loopsfrom the standard diagram of the relation, deleting any directed edge fromthe diagram if it joins two vertices that are also connected by a directedpath of two or more edges. Further if a is related to b we draw b higherup the page than a (and so we dont need an arrow on the edge)..

As an example we can draw both the standard diagram and the Hassediagram of on the set of subsets of {1, 2, 3}.

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3 Functions

3.1 Definition of a function, notation, examples

Definition 3.1 Given two sets X and Y , a function f from X to Y isa rule that, given any element x X defines an element f (x) in Y .3 things are necessary to define a particular function; the set X (calledthe domain), the set Y , (called the codomain), and the rule itself. Wecall f (x) the image of x under f .

In programming, the domain corresponds to the type of the input argumentof the function, and the codomain to the type of the returned value.We usually use letters like f, g, h as names of functions, or sometimesf1, f2, . . .. But it doesnt really matter.

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Commonly we describe a function using notation like

f : Z Z, f (x) = x2 or alternatively f : Z Z, x 7 x2

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My example above probably looks quite familiar, because the functiondeals with numbers, but theres no reason why either X or Y should bea set of numbers. They could be sets of anything.

The following all define valid functions.

X is the set of people in this room, Y = R, f (x) is the body temper-ature of x, for each x X .

X is the set of people in this room, Y the set of all female first names,f (x) is the first name of the mother of person x.

X is a set of records, Y the set of possible values of one particularcomponent, f (x) is the value of that component for record x.

Any relation R on sets A1, . . . An can be thought of as a functionfrom A1 . . . An to the set {True, False}. We define R(x1, . . . xn)to take the value True when the relation holds on (x1, . . . xn) and

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False otherwise. This explains why were using the same notation forfunctions as we already (sometimes) used for relations. A function withcodomain {True, False} is called a Boolean valued function.

The terms mapping and map mean exactly the same as function.

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For any set X , we can define a special function, which we call the identityfunction on X .

Definition 3.2 (Identity function) The identity function iX : X X . is defined by the rule i(x) = x for all x X .So iZ is a function with domain and codomain Z, iR a function withdomain and codomain R etc. Each identity function iX has the samerule: iX(x) = x for each x X .

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While the codomain of a function tells us where its output should lie (andso specifies its type) the range specifies the set of outputs completely.

Definition 3.3 (Range)Where f : X Y is a function, the set ofall elements of Y that are of the form f (x) for some x in X is called therange of f . In set theory, we would describe this set as

{y Y : y = f (x) for some x X}Some people use the word image instead of range.

For our first example f : Z Z, x 7 x2, the range of f is the set of allsquares, {0, 1, 4, 9, 16, 25, . . .}.

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3.2 The graphs of a function, directed and Cartesian

Since a function f is specified by the two sets X , Y and all the pairs(x, f (x)), it is actually a binary relation on X Y . Each element of Xis related by f to its image f (x) in Y (but careful, the boolean valuedfunction we associate with that binary relation is not f). So it is natural touse a diagram to represent the function, consisting of points and directedarcs. A cluster of points on the left of the diagram represents the elementsof X , another cluster of points on the right represents the elements of Y .For each element x of X , a directed arc joins it to its image f (x) in Y .

NB. If X and Y have some points in common we see each common pointtwice, once as an element of X , and once as an element of Y .

We call this diagram the directed graph of the function.

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Well look at some examples, not forgetting the identity function. It iseasy to identify the range of a function from its directed graph.

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Of course we are also used to representing functions by a different kind ofgraph, which uses axes and coordinates. The points of X are marked offon an x-axis, the points of Y on a y-axis, defining a coordinate sustem.Then each point (x, f (x)) is plotted. Well call this kind of graph theCartesian graph of the function.

Of course we can only use a Cartesian graph to represent a function thatmaps a set of numbers to a set of numbers. A directed graph can be useda bit more generally, but in practice were unlikely to use it if X and Yare infinite.

Well look at some examples, not forgetting the idenfity function. It iseasy to identify the range of a function from its Cartesian graph.

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3.3 Functions vs. non-functions

Let X be the set of student numbers of people in this room. Let Y1 bethe set of all female first names, let Y2 be the set of all first names.

If f1(x) is the first name of the person with student number x, then f1doesnt define a function from X to Y1, since some people in this roomare male. f1 : X Y1 is not a function because f1(x) is not defined asan element of Y1 for every x X ; i.e. f1(x) is outside the codomain forsome elements of X .

But the same rule f1 defines a function from X to Y2.

Now let f2(x) be the name of the friend of the person with student numberx. Wed like to use f2 to define a function from X to Y2, but we havea problem since some people have more than one friend, while some may

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have none. f2 : X Y2 is not a function because f2(x) is undefined orambiguous for some x in X .

For f : X Y to be a function we need exactly one value of f (x) foreach element x of X , and we need that value to be an element of Y .

Look at this pictorially. In the directed graph that represents a function,we see one directed arc out of each element of X , leading to an elementof Y . Some elements of Y may be at the end of more than one arc, andsome elements of Y may be at the end of no arc.

Here are some example of directed graphs, some of functions, some ofnon-functions.

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3.4 Composing functions

Sometimes we have more than one function and it makes sense to combinethem.

Definition 3.4 (Function composition) Given functions f : X Y and g : Z W ,if Y Z, the composite g f is the function with domain X andcodomain W defined by the rule

g f (x) = g(f (x)).If Y 6 Z, then g f is not defined.Note that we cant always compose two functions. We can only definethe composite g f of functions f and g if the codomain of f is in thedomain of g. We might have both of f g and g f defined, we might

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have either one but not the other defined, and we might have neitherdefined. Where both are defined they are probably not equal.

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Examples:

First let X = {1, 2, 3}, Y = {4, 5, 6, 7}, Z = {4, 5, 6, 7, 8}, W ={1, 2, 3, 4, 5} and define f : X Y, g : Z W by f (1) = 4,f (2) = 6,f (3) = 7, g(4) = 2, g(5) = 3, g(6) = 4, g(7) = 5, g(8) = 5.Draw the directed graphs of f and g and study the composition graphically.

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Next let R0 = {x R : x 0}, and define f : R R0 by f (x) =x2, g : R R by g(x) = x 1, h : R0 R0 by h(x) = x + 1.Then compute the compositions (and see that h g doesnt work).

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3.5 Inverse functions

Sometimes we want to undo the effect of a function. We know f (x) butwe want to know where it came from, what x was. That could be veryimportant if we wanted to detect an error in a program. Maybe we havea program that runs through a large set of input values x, and for eachvalue of x we have an output value y, computed using some function f .If the program is interrupted we may want to recover x. Maybe memoryconstraints meant we couldnt store it. Can we recover x?

We can if f has an inverse.

Definition 3.5 (Inverse function) Given a function f : X Y ,then a function g : Y X is an inverse for f if g f (x) = x for allx X and also f g(y) = y for all y Y .

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Another way of phrasing the above is to say that gf = iX and fg = iY .We say above that g is an inverse for f . In fact if f has an inverse, itonly has one. But it doesnt have to have one at all. If f has an inversewe usually call it f1.To explore, well draw the graphs (directed or Cartesian) of some functions.If the function has an inverse well draw its graph too.

f1 : R R, x 7 2x.

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f2 : R R0, x 7 x2. (We define R0 to be {x R : x 0}.)

:f3 : Z Z : x 7 2x.

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f4 : {0, 1, 2, 3, 4, 5} {0, 1, 2, 3, 4, 5} : x 7 5 x.

f5 : R R : x 7 5 2x.

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Where f is invertible, we get the directed graph of f1 by reflecting thedirected graph of f left to right, and we get the cartesian graph of f1by reflecting the directed graph of f in the line y = x.

If f isnt invertible the reflections of its graphs wont be the graphs offunctions.

To illustrate this, well draw the reflected graphs of each of our examplesthat dont have inversess:

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3.6 Properties that make a function invertible

We look at our 5 examples to see what can stop a function having aninverse; f2 and f3 were not invertible.

We have problems defining an inverse for f2 because each of the valuesin the codomain is the image of two different values in the domain. Wedont know which value the inverse should take it back to. E.g. f2(2) =f2(2) = 4, but we cannot have both f12 (4) = 2 and f12 (4) = 2.We have problems defining an inverse for f3 because there are values ofthe codomain that are not in the range. We cannot define f13 (3) because3 is not in the range.

These problems are visible in the graphs of f2, f3. What we have observedis that f2 is not injective, and f3 is not surjective.

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Definition 3.6 (Injective, surjective, bijective) Let f : X Ybe a function.

f is injective (or 1-1) if whenever x1, x2 are distinct elements of X , thenf (x1) 6= f (x2).f is surjective (or onto) if every element y Y is an image f (x) for atleast one x X .f is bijective if it is both injective and surjective.

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An injective mapping is called an injection, a surjective mapping a surjec-tion, a bijective mapping a bijection.

Let f : X Y be a function. It turns out that if f is injective, then there is a function g : Y X for which gf = iX .g is called a left inverse for f . if f is surjective, then there is a function h : Y X for whichf h = iY . h is called a right inverse for f . Actually there could be many left or right inverses. if f is bijective, then f actually has an inverse, which is both a leftinverse and a right inverse.

So the properties of injectivity and surjectivity together are what we needto have an inverse.

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We can check these properties easily on the graphs of functions.

In the Cartesian graph: for an injective function we see at most one point with any given y-coordinate, for a surjective function we see at least one point with any given y-coordinate (for y in the codomain).

In the directed graph: for an injective function we never see two arrows to the same point inthe codomain, for a surjective function we see at least one arrow to each point in thecodomain.

We can draw some examples of graphs, and recognise which of them rep-resent injective functions, which surjective and which bijective functions.

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Examples (Cartesian graphs):

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Examples (directed graphs):

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3.7 Computing an inverse for a function

If we want to try and compute an inverse for a function, we start by lookingfor a left inverse.. Then we check to see if it is also a right inverse.

e.g. As an example, we compute the left inverse of

f : R R, x 7 2x + 3We rearrange the equation y = 2x + 3 to get x on its own. In this waywe get a function g : R R for which g(y) = x, so g f (x) = x.

Now we can check that f g(y) = y for all y R.91

3.8 Partial functions

Definition 3.7 Given two sets X and Y , a partial function f from Xto Y is a rule that defines an element f (x) Y for some elements x ofX , but may be undefined for others.

We write f : X 6 Y to indicate that f is a partial function but not afunction from X to Y .

The subset of X consisting of those x for which f (x) is defined is calledthe domain of f .

We sometimes use the term total function instead of just function, if wewant to stress the fact that we have a function, not just a partial function.

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If f : X 6 Y is a partial function with domain Z, we can define a (total)function g : Z Y by the rule g(z) = f (z) for all z Z. We call g therestriction of f to Z, and write

g = f |Z.Since every partial function has a total function closely associated with it,a lot of what we have said already about functions remains true for partialfunctions.

A partial function isnt much less than a function, and sometimes its goodenough for our purposes.

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Examples of partial functions:

f from R to R defined byf (x) =

(x),

wherever this makes sense. This is a partial function with domain {x R : x 0}. g from R to R defined by

g(x) =1

x2 1wherever this makes sense. This is a partial function with domain R \{1,1}.

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We can find a partial function g from Z to Z that is a left inverse ofthe function

f : Z Z, x 7 2x + 1by rearranging y = 2x + 1 to get x on its own.

We get

x =y 12

The reason that

g : Z 6 Z, y 7 y 12

is not a total function is that y12 is not an integer for all y Z. Thedomain of g is the odd integers; it is equal to the range of f .

The partial function g satisfies g f = iZ.

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Using g its now easy to construct a (total) function that is a left inverseof f . We simply define h(y) = g(y) whenever g is defined, and h(y) = 0otherwise. Our construction guarantees that h f = iZ. So h;Z Z isa left inverse of f .

But f does not have an inverse. It cannot. It is not surjective.

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3.9 Functions defined recursively

The rule that defines a function is not always given explicitly. Sometimesit could be given explicitly, but can be given much more simply in anotherform, which describes the value of the function at each value of x interms of its value at smaller values of x. In many examples, the functionis defined on the positive integers, and its value at x can be defined interms of its value at positive integers less than x.

e.g. The factorial function, fact(n), also written n!. This is a functionfrom N to N, which is defined for any positive integer n to be the productn(n 1) . . . 1. Its defined very simply recursively by the pair of rules

fact(n) = n fact(n 1)and

fact(1) = 1

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So the easiest way to program it is as follows (in C):-

int factorial(int n) /* n assumed >0 */

{

if (n==1 ) return 1;

return n*factorial(n-1);

}

e.g. The function f : N N defined by f (n) = 2n can be definedrecursively by the rules

f (1) = 1, f (n) = 2f (n 1)and similarly easily programmed.

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A recursive definition of a function always has two components, the recur-sive rule (which defines f (n) in terms of some of the earlier values of f)and initial conditions, which fix the first few values of f , f (1) (or f (0)),and possibly f (2), f (3), . . .. The number of initial conditions that arenecessary depends on the type of recursion. If the recursion is one step(that is, the rule defines f (n) in terms of f (n 1) only, then only oneinitial condition is necessary. But if the recursion is k-step, defining f (n)in terms of f (n 1), . . . f (n k), then k initial conditions are necessary.Examples 3.8Where f : N N is the function given recursively bythe rule f (n) = 2f (n 1) and f (1) = 2, we havef (2) = 2 2 = 4, f (3) = 2 4 = 8,f (4) = 2 8 = 16. In fact thefunction can be described very simply by the rule f (n) = 2n.

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Where f : N N is the function given recursively by the rule f (n) =2nf (n 1) and f (1) = 2, we have

f (2) = 2 2 2 = 8.f (3) = 2 3 8 = 48.f (4) = 2 4 48 = 384

This function is in fact 2nfact(n) (also written 2nn!).

Where f : N N is given by f (n) = 12 3n, we havef (n)

f (n 1) =12 3n

12 3n1 = 3

So f (n) = 3f (n 1). f can be described recursively by the rule f (n) =3f (n 1) and the initial condition f (1) = 36.We can compute further examples.

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4 Graphs

4.1 Definition and examples, notation

Definition 4.1 (directed graph) A directed graph consists of a set Vof vertices, and a set E of ordered pairs of vertices, called directed edges.

The directed edge e = (x, y) joins vertex x to vertex y; x is called theinitial vertex, y the terminal vertex of e.

If the two vertices of e coincide, then e is called a loop. Two distinctedges cannot share both their initial and their end vertices.

We always think of directed graphs pictorially, representing each vertex asa point in a plane, a directed edge (x, y) as a directed arc from the pointx to the point y. We often abbreviate directed graph to digraph.

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We understand that the same graph may be displayed on the page inmany different ways. It is defined simply by its set of vertices and theconnections between them, and it is irrelevant how these are placed onthe paper.

Hence the first two diagrams below represent the same directed graph.Strictly speaking the third diagram represents a distinct (but isomorphic)graph, since its vertices have been differently labelled, but we dont alwaysneed to label the vertices of a graph.

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We have already met directed graphs, used as diagrams for both relationsand functions (see sections 2 and 3); we can learn a lot about relationsand functions by studying the properties of the associated directed graphs.

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Sometimes we prefer not to have arrows on our edges, and to outlawloops.

Definition 4.2 (graph) A graph consists of a set V of vertices, and aset E of edges each of which is a set of two distinct vertices of V . Wheree = {x, y} is an edge, we say that e joins x and y..Of course we always think of graphs pictorially too, representing verticesas points in a plane and edges as undirected arcs between pairs of vertices.

We have already met examples of graphs as the equivalence graphs ofequivalence relations (see section 2).

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Given any digraph we can form a graph by deleting any loops and con-necting by an edge any two vertices connected by 1 or 2 directed edges,e.g.

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We define the complete graph on n vertices Kn to be the graph on nvertices in which any two vertices are joined by an edge. Kn has n(n1)/2edges.

Examples:

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We define the cycle graph Cn to be the graph consisting of n verticesjoined together in a single cycle of n edges.

Examples:

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4.2 Paths, connectivity, components

Definition 4.3We define a path (of length n) in a graph or directedgraph to be a sequence of vertices v0, v1, . . . vn such that for any i,(vi, vi+1) is a edge or directed edge. Such a path is called a circuit(of length n) if v0 = vn, and a cycle if v0 = vn and all of v0, . . . vn1are distinct. The length is the number of edges in the path/circuit. Agraph is said to be connected if any two vertices x, y are connected by apath; a directed graph is said to be connected if its underlying graph isconnected.

Examples:

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Definition 4.4 Suppose that G = (V,E) is a graph (or directed graph).

A subgraph of G is a graph (or directed graph) G = (V , E) whosevertex set V is a subset of V , and whose edge set E is a subset of E.A component.of G is a connected subgraph G = (V , E) of G such thatno edge of G joins a vertex in V to a vertex outside V .

Examples.

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For the equivalence graph of an equivalence relation, the componentsof the graph define the equivalence classes of the relation; two verticesare in the same component if and only if they are related.

Example, the equivalence graph of 3 on {0,1,2,3,4,5,6,7,8,9}:

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4.3 Bipartite graphs

Definition 4.5 A graph (or directed graph) is bipartite if its vertex setV can be written as a disjoint union of two subsets V1 and V2 in sucha way that any edge in the graph joins a vertex in V1 to a vertex in V2.More generally a graph (or directed graph) is k-partite if its vertex set Vcan be written as a disjoint union of k subsets V1, . . . , Vk in such a waythat every edge joins two vertices in different subsets.

An alternative way of phrasing this is to say that a graph is k-partite ifthe vertices can be coloured with k colours in such a way that any edgejoins two vertices of different colours. Of course if a graph is k-partite itsalso (k + 1)-partite.

We can draw lots of examples.

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First some bipartite graphs (Note that the directed graph of a function isalways bipartite.)

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Next some tripartite graphs

And some other multipartite graphs.

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We define the complete bipartite graph Kr,s to be the bipartite graph onr+ s vertices with rs edges each joining a vertex in a subset of r verticesto one of the other s vertices.

We define complete k-partite graphs similarly, e.g. K2,3,4

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It can be proved that any graph that can be drawn on a piece of paperwith no edges crossing can be coloured with 4 colours (and so is 4-partite).This is the famous four colour theorem.

Another well known theorem in graph theory tells us that a graph canbe drawn on a piece of paper with no edges crossing provided it doesntcontain a subgroup that deforms to give either K5 or K3,3.

It is easy to check whether or not a graph is bipartite (2-partite). A graphis bipartite if and only if all its cycles have even length.

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4.4 Trees and forests

In computer science, certain graphs, known as trees are important.

Definition 4.6 A tree is a connected graph without any cycles. It mayor not have directions assigned to its edges. A vertex with just one edgethrough it is called a leaf. Sometimes one vertex is selected to be calledthe root of the tree; then the tree is known as a rooted tree. In that case,the tree is called a binary tree if every non-leaf is joined to 2 vertices thatare further from the root than it. If x is a vertex in a rooted tree, thenthe vertex joined to x and nearer to the root than x is called its parent,and the other vertices joined to x (and further from the root than x) arecalled its children.

A forest is a graph without any cycles, that is a graph whose componentsare trees.

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Some diagrams of examples.

119

120

Rooted trees are used to define directory stucture.

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4.5 Parsing trees

Trees are also used for parsing, e.g. of arithmetic expressions, such as

(x+x(y+z))+y(xz) or[x (x y)3 + (x + y) (x y)

]2.

We see , + and as binary operations, and exponentiation by n (foreach integer n) as a unary operation. Then any arithmetic expression isbuilt up using a combination of those binary and unary operations on a setof constants and variables. We parse such an expression using a rootedtree in which each vertex has either one or two children.

We form a parsing tree as follows. Each leaf is labelled by a constantor variable, and each non-leaf by an operation. A non-leaf is then alsoassociated with the sub-expression that is formed by applying the operation

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that labels that vertex to the one or two (maximal) subtrees whose root(s)is/are the children of that vertex.

NB. Note that when we parse an arithmetic expression, we always parsex+ y z as x+ (y z), x+ y3 as x+ (y3), and x y3 as x (y3), etc.That is, exponentiation has priority over multiplication and addition, andmultiplication has priority over addition.

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Examples:

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4.6 Spanning trees

Definition 4.7 A spanning tree for a graph is a subgraph, involving allthe vertices, that is also a tree.

Given a graph G = (V,E) we can easily construct a spanning tree asfollows.

If G is not a tree, it contains a cycle. So we look for an edge that is in acycle, and delete it, to get a new graph G = (V,E) with all the verticesof G, but one fewer edge. If G is a tree, then it is a spanning tree for G.If not, it contains a cycle, and we repeat the above procedure.

We continue until we arrive at a tree.

Example:

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5 Recursion and induction

5.1 Examples of some sets defined inductively

Z+ is defined as follows:-(1) 1 Z+; (2) If x Z+ then x + 1 Z+.Nothing is in Z+ that is not forced by either (1) or (2) to be in Z+.

The set P of polynomials in x with real number coefficients is definedas follows:-

(1) For any c R, c is in P (2) If f is in P then so are xf and f + c,for any c R.Nothing is in P that is not forced by (1) or (2) to be in P .

The set T of binary trees can be defined as follows:-

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(1) The graph K1 is in T . (2) If T is a binary tree and v is any leaf ofT , then the graph formed from v by attaching two edges and two newvertices below v is in T . (We can draw a picture to illustrate this.)

Nothing is in T that is not forced by either (1) or (2) to be in T .

What are the advantages of recognising that a set can be defined induc-tively?

We can give recursive definitions for functions that have that set astheir domain. Such definitions often give rise to very straightforwardprogramming, e.g. look at the way we could program the factorial

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function on the positive integers.

We can prove things about the set using the principle of mathematicalinduction, e.g. we can prove:-

The sum of the first n positive integers (i.e. 1 + 2 + 3 + . . . + n) isn(n + 1)

2.

If T is a binary tree with n leaves then T has 2(n 1) edges.How do we prove something using the principle of mathematical induction?

We split the argument into two pieces, the base of the induction and theinductive step. The base of the induction deals with the first elementof the set, defined by the first rule, and the inductive step deals with thesecond rule.

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5.2 Some examples of proof by induction

Example 5.1 To prove that the sum of the first n positive integers (i.e.1 + 2 + 3 + . . . + n) is

n(n + 1)

2

Proof:

Base of the induction We prove the statement for n = 1. The sum of thefirst 1 positive integers is just the first positive integer, i.e. 1 itself. Theformula n(n + 1)/2 takes the value 1(1 + 1)/2 = 1 when we put n = 1into it. So the formula gives the correct answer when n = 1. Hence thebase of the induction is proved.

The inductive step Suppose that the sum of the first n positive integers

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is given by the formulan(n + 1)

2To prove the inductive step we need to deduce from this that the sum ofthe first n + 1 positive integers is given by the formula

(n + 1)((n + 1) + 1)

2=

(n + 1)(n + 2)

2.

Now the sum of the first n + 1 positive integers is

1 + 2 + . . . + n + (n + 1) = (1 + 2 + . . . + n) + (n + 1)

and since we know the formula for the first n we can write this as

n(n + 1)

2+ (n + 1)

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When we tidy this up we get

(n + 1)(n2+ 1

)= (n + 1)

(n + 2)

2

i.e. exactly the formula we were looking for. Hence the inductive step isproved.

The principle of mathematical induction says that if we check just thesetwo things we have proved that the formula holds for every positive integer.2

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Example 5.2 If T is a binary tree with n leaves then T has 2(n 1)edges.

Proof:

Base of the induction We need to prove the result for the binary tree withjust one leaf, i.e. for the tree that is a single vertex. This tree has noedges. And this fits with the formula, since 2(1 1) = 0.So the base of the induction is proved.

The inductive step Now suppose that we know that the result holds forbinary trees with n leaves. We want to deduce from this that the resultholds for binary trees with n + 1 leaves.

Now suppose that T is a binary tree with n + 1 leaves. Then we canfind two leaves in T that are joined by edges to a single vertex v; whenthe two vertices and two edges are deleted from T we get a binary tree

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T with 2 fewer edges and 2 fewer vertices than T . That is, T is formedfrom T by adding two edges and two vertices below its leaf T .

(as in diagram)

The vertex v was a leaf of T but is not a leaf in T , but every otherleaf of T is also a leaf of T . So T has n leaves. Then since the resultholds for binary trees with n leaves, T must have 2(n 1) edges.Now since T has two more edges than T , the number of edges in T is

2(n 1) + 2 = 2n = 2((n + 1) 1).So the inductive step is proved.

The principle of mathematical induction says that by checking just thesetwo things we have proved that the formula holds for every binary tree. 2

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Example 5.3 2n n3 for any integer n with n 10NB: to do this example, we need to know the expansion

(n + 1)3 = n3 + 3n2 + 3n + 1

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Example 5.4 n3 n is divisible by 3 for any integer n.

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6 Propositional logic

6.1 Statements, propositions, paradoxes

This point of this section is to set up a framework for reasoning by ma-nipulating statements, in particular special kinds of statements that arecalled propositions.

Definition 6.1 (statements, propositions) A statement is a sentencethat is not a question or a command. A proposition is a statement thatis either true or false, but not both. If a proposition is true we say it hastruth-value True. Otherwise it has truth-value False.

In programming, we usually use 1 to represent True and 0 to representFalse. In this course well often use T and F .

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Examples 6.2

1. Some true propositions:

(a) 2 + 3 = 5. (b) Mumbai is in India. (c) Spurs won the FA cup in1991.

2. Some false propositions.

(a) 10 > 87. (b) Newcastle is south of Leeds.

3. Propositions whose truth-value depends on one or more variables -open statements.

(a) n is an even integer. (depends on n). (b) She is taller than me.(Depends on she and me). (c) x + y > 7. (Depends on x and y).

4. The following are NOT propositions or statements.

(a) Well done! (b) How did you manage that?

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5. The sentence: This sentence is false. is a statement, but not aproposition. It cannot be true, and it cannot be false. It is in fact aparadox.

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6.2 Compound propositions, logical connectives

Definition 6.3We make compound propositions by glueing togethersimple propositions using the logical connectives and, or and not.

So from the two simple propositions

Roses are always red. Violets are always blue.

we can make various compound propositions

Roses are always red and violets are always blue.Roses are always red or violets are always blue. Roses are not always

red.

The truth-value of a compound proposition is determined both by thetruth-value of the constituent propositions, and by the way in which they

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are connected together. Well learn how to manipulate compound propo-sitions; the language and laws of this manipulation are known as thepropositional calculus.

To stop our explanation of this getting too wordy, well use symbols ratherthan the English language.

Well use letters like p, q, r to stand for propositions and call them propositionalvariables, (called conjunction) means and, (disjunction) means or, and means not. T is a true proposition, F a false proposition,We call an expression that we build out of these symbols a propositional formula

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Strict rules define the way in which and, or and not should be inter-preted in logic (not necessarily as you might expect in English where themeanings are not always consistent).

Given propositional formulae f, g,

f g is defined to be true if both f is true and g is true, and is definedto be false otherwise.

f g is defined to be true if f is true or g is true, including when bothare true, and false otherwise.

f is defined to be true when f is false, and false when f is true.Note the meaning of as or is defined to be inclusive, as in Would youlike milk or sugar with your tea? rather than exclusive as in Would youlike tea or coffee?.

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6.3 Logical equivalence, truth tables

Definition 6.4 (Logical equivalence) .Two propositional formulae, f, g are said to be logically equivalent if theformula f is true whenever the formula g is true, and vice versa. We writef g.e,g, for any p, q, r, p (q r) (p q) (p r)

We use truth tables to test for logical equivalence between propositionalformulae. Truth tables play the same role in logic that Venn diagrams doin set theory.

The truth table of a formula has many rows and columns. Theres a column for each variable in the formula, and then one for eachappearance of each of the (binary or unary) operations in the formula.

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Theres a row for each possible combination of values for the variablesin the formula.An iterative process (to be explained) computes the value of the wholeformula, and inserts it into the appropriate column of the truth table.Two formulae are logically equivalent if and only if their truth tablesmatch, that is, if the values of the formulae match as the variables runthrough all possible sets of values.

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6.4 Construction of truth tables

The iterative process constructs the truth table for a formula by parsingthe formula, and then combining the truth tables for the basic operationsinto which it breaks down.

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Truth tables for the basic operations

The building blocks for any truth table are the truth tables for the threebasic operations, , and . These tables are as follows:-

p q p qT T TT F FF T FF F F

p q p qT T TT F TF T TF F F

p pT FF T

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Parsing the formula

When we have a formula involving more than one operation we need tostart by parsing it. In this way the operation at the root of the parsing treeis identified. Each other operation is identified as the root of a subtree,and hence corresponds to a subformula.

Within the truth table we have first a column for each variable, and thena column for each operation appearing in the parsing tree (each operationheading as many columns as it labels vertices of the parsing tree). Weuse a column headed by an operation to store the value of the appropriatesubformula. We build up the value of the whole formula, working fromthe leaves of the parsing tree until we reach the root.

Well illustrate this process with an example.

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Example 6.5 The truth table for (p q) (p r)First we draw the parsing tree.

The root is labelled , and the two non-leaves are labelled , corre-sponding to subformulae p q and p r. So we have a truth table with6 columns, 3 labelled by the variables p, q, r and then 3 more, labelledp q, and p r, where we store the values of the subformulae p q,the whole formula and the subformula p r.

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p q r (p q) (p r)T T TT T FT F TT F FF T TF T FF F TF F F

The values of the whole formula are in the 5th column.

NB: We should always run through all possible values of the variables ina systematic order (as we have done here), to make it easier to comparetruth tables of different formulae to show their logical equivalence.

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Example 6.6We prove using truth tables that p (q r) (p q) (p r)We just built the truth table for . (p q) (p r) Now we build the truthtable for p (q r).

p q r p (q r)

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We find the truth value for the whole formula in the fourth column of thistable, and compare it with the calues in the fifth column of the other table.Since the entries in those columns match (and the values of p, q, r havebeen enumerated in the same orders), we have proved the equivalence ofthe two formulae.

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6.5 Rules relating the operations of propositional logic

The equivalence weve just proved should look familiar; weve seen thesame equivalence for sets. In fact, using truth tables, we can also verifyall of the following, for propositional formulae f, g, hs.

Commutative laws. f g g f f g g fAssociative laws. f (g h) (f g) h) f (g h) (f g) hDistributive laws. f (g h) f (g h)

(f g) (f h) (f g) (f h)f T f f F ff f F f f T

De Morgans laws. (f g) (f ) (g) (f g) (f ) (g)Idempotent laws. f f f f f f

(f ) = f154

Its clear that theres a connection between set theory and propositionalcalculus. Its like two different languages. Sets translate to propositionalformulae, to , to , U to T and to F . Theres a reason for this,which will become clearer later in the course.

We see again the duality that we saw with sets, where is paired with, and T with F .

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6.6 Tautologies and contradictions

Definition 6.7 (Tautology, contradiction) .A propositional formula that is always true is called a tautology. A propo-sitional formula that is always false is called a contradiction.

Example 6.8 Show that ((p q)) (p) is a tautology.We simply construct the truth table. Since all values of p and q give a Tin the column for the whole formula (under the , it is a tautology.

p q ( (p q)) (p)

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6.7 Translating English sentences into the language of propo-sitional logic

We need to be able to break a proposition down into its constituent parts.We have to think very hard about what the sentence really means. Welllook at some examples:

Sue is a beautiful woman can be broken down as Sue is beautiful andSue is a woman. The negative of this is then equivalent to Either Sueis not beautiful or Sue is not a woman. (Remember de Morgans law.)

Mike is a wonderful guitarist is already as simple as it can get, i.e. itdoes not mean the same as Mike is wonderful and Mike is a guitarist.Nobody said that Mike was wonderful.

To translate The work is easy to do but not satisfying we first translatebut as and not. Then the whole sentence can be broken down as

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The work is easy to do and the work is not satisfying. The negative ofthis is equivalent to Either the work is not easy to do or it is satisfying.Soon we shall see that this negative is equivalent to the statement Ifthe work is easy to do then it is satisfying.

We can build further propositional formulae with the introduction of ad-ditional connectives.

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6.8 The connectives , and

We define connectives , and by specifying their truth tables, asfollows:-

p q p qT T TT F FF T TF F T

p q p qT T TT F TF T FF F T

p q p qT T TT F FF T FF F T

We can verify, using truth tables that

q p q,p q q p,p q (p q p q).

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We read f g as f implies g, or if f then g . Similarly we readf g as f is implied by g and f g as f if and only if g,Note that by definition f g is true provided that whenever f is truethen g is also true. Hence in logic, f g is true whenever f is false,irrespective of the value of g. This may not seem consistent with Englishusage, But it is true in logic. Hence the proposition If pigs can fly thenthe pope is a Protestant is true.

Note that the negative of p q is equivalent to the negative of p q,and hence by de Morgans law is equivalent to pq. Hence the negativeof If pigs can fly then the pope is a Protestant must be equivalent toPigs can fly and the pope is not a Protestant.

NB: weve used the notation , and . Often you see , and , meaning the same.

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7 Predicate logic

Weve already spent a little time studying propositional logic, manipulatingformulae consisting of variables representing propositions linked by thelogical connectives , , , , and . In propositional logic, wework from the principle that a proposition is either true or false, and weare interested in knowing which truth values of the propositional variablesinvolved in a formula make the compound proposition it represents true,and which make it false.

But propositional logic can be a bit restrictive and doesnt allow verysophisticated arguments. For more sophistication we need to allow ma-nipulation of formulae at a level lower than propositions. We need tobreak each proposition down into its constituent parts, such as variables,constants, logical connectives and predicates. Predicate logic allows us to

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do this.

To illustrate this well explain how predicate calculus allows us to deducethe statement

(3):My cat has blue eyes

from the pair of statements

(1):My cat is a Siamese cat and (2):Every Siamese cat hasblue eyes,

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We need to write each of the three statements in terms of predicates(relations)..

We can write the first statement in terms of the binary predicate .Setting its two arguments equal to constants my cat and (the set)Siamese cats we get a translation of the first statement as

(1):my cat Siamese cats.We define a unary predicate Q to mean has blue eyes. Then the thirdstatement translates as

(3):Q(my cat)

The second statement is a little more complicated. We can express it usingthe two predicates and Q, and a little more. In fact we can phrase it as

(2):For every x Siamese cats, Q(x).164

Now the logic should be clear. The second statement tells us that Q(x)holds whenever x is an element of the set Siamese cats. The firststatement tells us that my cat is in that set. Hence Q(my cat) holds.Predicate calculus gives us a mechanism (including symbols) to expressthe second statement properly, and hence to reason in this way.

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7.1 The universal quantifier

We use the symbol in predicate logic to mean for all, for every,for each. x means for all x. Often for all x really means for allappropriate x, that is, for all x in some set of values which make sensein the current context. In that case we write x S, where S is that setof appropriate values. the symbol is known as the universal quantifier.Hence the statement above can be written in symbols as

x Siamese catsQ(x)Note that although x appears as a variable in this proposition, it is not free,that is, the proposition does not make sense if we replace x by a constantWe say that x is a bound variable. Although x cannot be replaced bya constant, it can be replaced by another variable, without changing the

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value of the proposition. The proposition

y Siamese catsQ(y) and x Siamese catsQ(x)are logically equivalent to each other.

By contrast the variables x in the two statements x Siamese catsand Q(x) are free; the statements are meaningful when those variablesare replaced by constants (as in the first and third statements).

In order to be able to manipulate formulae involving the universal quanti-fier, we need to know precisely when a formula involving it is considered(in logic) to be true. Given a set of variables S and a unary predicate P ,the proposition

x S P (x)is defined to be true precisely if the proposition P (x) is true for everyelement x of S. Otherwise the proposition is false.

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Similarly the propositionxP (x)

is defined to be true precisely if P (x) is true for all possible values of x(whatever that means).

Examples 7.1

Where, for x N, the statement P (x) means x is a perfect square,(that is, x = y2 for some integer y) the statements P (0) and P (1) aretrue, P (2) is false, P (3) is false, P (4) is true . . ..So the statement x NP (x) is false.Where P (x)means x loves the underdog, the statement x, P (x)meanseverybody loves the underdog.

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7.2 The existential quantifier

We use the symbol in predicate logic to mean there exists, there is,for some. x means there exists an x. We may use x or x S,for some set S. Just like , the existential quantifier has the effect ofturning a free variable into a bound variable.

Given a set of variables S and a unary predicate P , the proposition

x S P (x)is defined to be true precisely if the proposition P (x) is true for at leastone element x of S. The proposition is defined to be false if P (x) is falsefor every element x of S.

Similarly the propositionxP (x)

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is defined to be true precisely if P (x) is true for at least one value of x.

Examples 7.2

Where, for x N, P (x) means x is a perfect square, since P (0) is true,the statement x NP (x) is true.Where P (x) means x loves the underdog, x, P (x) means somebodyloves the underdog.

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7.3 Multiple quantifiers

It is perfectly possible to apply a (universal or existential) quantifier toa predicate involving several variables, or even to apply more than onequantifier. Each quantifier binds one variable, and of course two differentquantifiers cannot simultaneously bind the same variable.

e.g. We can write P (x) above as y Zx = y2, and so the (false)statement x NP (x) can be written as x Ny Zx = y2.

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7.4 Relating the existential and universal quantifiers

Notice that, for any unary predicate P , and for any set S, the followingtwo identities hold.

(x S P (x)) x S (P (x))(x S P (x)) x S (P (x))

Examples 7.3

Where P (x) is, as above x is a perfect square. x N (P (x)) trans-lates as there is a natural number which is not a perfect square.

We know this to be true, and we used this fact to prove the statementx N, P (x) to be false.x NP (x) is true.

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7.5 Translating English sentences using quantifiers

The square of any integer is positive. translates as

x Z x2 0The sum of an odd integer and an even integer is always an odd integer.translates as

x OddInt y EvenInt x + y OddIntBetween any two odd integers there is an even integer translates as

x OddInt y OddInt (x < y (z EvenInt x < z < y))or as

i Z j Z (2i+1 < 2j+1 (k Z 2i+1 < 2k < 2j+1))

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The principle of mathematical induction uses the fact that the followingis a tautology:-

(P (1) (n NP (n) P (n + 1))) (n NP (n))

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7.6 A systematic approach to translation

Lets try and be a bit more systematic, and a bit more ambitious. Id liketo be able to express some famous results about natural numbers usingpredicate calculus:

The product of two odd integers is odd. The square root of 2 is irrational. (Hippasus, 600 BC). The only integer solutions to xn + yn = zn with n > 1 occur whenn = 2 (Fermats last theorem).

Every natural number can be written as the sum of four squares (dueto Lagrange 1770)

An odd prime can be written as the sum of two squares if and only ifit is congruent to 1 mod 4. (Fermat)

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7.7 First we find expressions for the components

Suppose that n N.n is odd if k Z(n = 2k + 1)n is even if k Z(n = 2k)n is composite if r, s Z+ \ {1}(n = rs)n is prime if (r, s Z+ \ {1}(n = rs)) or, alternatively, r, s

Z+ \ {1}(n 6= rs))n is congruent to 1 mod 4 if k Z(n = 4k + 1)n is a perfect square if k Z(n = k2)n is a sum of two squares if k, l Z(n = k2 + l2)

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7.8 Quantifying more than one variable

Note that r, s is shorthand for r,s and r, s is shorthand for r,s,and that its legitimate to use this shorthand, since r,s means exactlythe same as s,r, and r,s means exactly the same as s,r.However when we mix two different quantifiers, the order of compositionis important, that is r,s and s,r do not mean the same.

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7.9 Translating whole statements

In general translation between logic and English (in either direction) iseasiest done in stages. We parse the statement into components, andthen translate each component. We can translate The product of twoodd integers is odd first as

For any m,n, if m and n are odd integers, then mn is an odd integer,

and then use the translation we know for n is an odd integer.

In this way we arrive at

m,n Z((i, j Z(m = 2i+1n = 2j+1)) (k Z(mn = 2k+1)))Similarly, The product of two even numbers is even translates as

m,n Z(i, j Z(m = 2i n = 2j) (k Z(mn = 2k)))178

In the other direction, the statement

m,n Z(k Z(mn = 2k) ((i Z(m = 2i)(j Z(n = 2j))translates to English first as

For any m,n, if mn is an even integer then either m is an even integeror n is an even integer,

and then as

If a product of two integers is even then one of the two factors must beeven

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The square root of 2 is irrational translates as

(p, q Z+(2q2 = p2))Fermats last theorem translates as

x, y, z, n N, (xn + yn = zn n 2)Lagranges theorem about sums of squares translates as

n N(i, j, k, l N(n = i2 + j2 + k2 + l2))

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Fermats theorem about sums of 2 squares is a little harder to express.

p is an odd prime translates as

(p Z p 6= 2 r, s Z+ \ {1}(n 6= rs))while p is a sum of two squares translates as

k Z,l Z(p = k2 + l2)We can translate For any p, if p is a prime congruent to 1 mod 4, thenp is the sum of two squares as

p Z+((k(p = 4k + 1) (r, s Z+ \ {1}(p 6= rs)))

(l,m Z(p = l2 +m2))).(We can leave out p 6= 2 since 2 is not of the form 4k + 1).

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We can translate For any positive integer p, if p is an odd prime and thesum of two squares, then p is congruent to 1 mod 4 as

p Z+((p 6= 2) (r, s Z+ \ {1}(p 6= rs)) (l,m Z(p = l2 +m2)))

((k Z(p = 4k + 1)))The theorem is the and of those two statement.

Unfortunately it is not logically equivalent to the statement

For any p, p is a prime congruent to 1 mod 4, if and only if p is the sumof two squares,

nor is it logically equivalent to the statement

For any p, p is an odd prime and the sum of two squares if and only if pis congruent to 1 mod 4.

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Neither of those last two statements is true.

For 20 = 22 + 42 is the sum of two squares, but is not prime, and is notcongruent to 1 mod 4.

And 9 = 4 2 + 1 is congruent to 1 mod 4, but it is not prime (it is thesum of two squares, 32+02). 21 = 10 2+ 1 is also congruent to 1 mod4. Its not prime, and it is not the sum of two squares.

The point is that we need (on both sides of the implication) to state thatp is prime.

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Maybe the simplest way to express the theorem is as follows:

For any positive integer p, p is an odd prime and the sum of two squaresif and only if p is a prime and is congruent to 1 mod 4.

That is,

p Z+(((p 6= 2) (r, s Z+ \ {1}(p 6= rs)) (l,m Z(p = l2 +m2)))

((r, s Z+ \ {1}(p 6= rs)) ((k Z(p = 4k + 1)))))

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8 Boolean algebra

8.1 Definition of a Boolean algebra

A Boolean algebra is a set S, containing 2 special elements the zero (0)and unity (1), equipped with operations of addition (+), multiplication() (both between pairs of elements) and complementation () (of singleelements), and satisfying the following axioms.

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Commutative laws: x y = y x x + y = y + xAssociative laws: x (y z) x + (y + z)

= (x y) z = (x + y) + zDistributive laws: x (y + z) x + (y z)

= (x y) + (x z) = (x + y) (x + z)Behaviour of 1 and 0: x 1 = x x + 0 = xBehaviour of complements: x x = 0 x + x = 1The symbols weve used, i.e. 0, 1,+, ,, arent a part of the definition.We could just as easily use other symbols.

Weve already met examples of Boolean algebras in this course.

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8.2 Examples of Boolean algebras

For any set U , the set of all subsets of U is a Boolean algebra, forwhich multiplication is given by , addition by , complementation byc, 1 by U , and 0 by .

For any set (say {p, q, r} ) of propositional variables, the set of allpropositional formulae in these variables is a Boolean algebra, for whichmultiplication is given by , addition by , complementation by , thezero by FALSE and the unity by TRUE.

The set {0, 1} forms a Boolean algebra, in which 0 is the zero and 1the unity when we define

0 + 0 = 0, 1 + 1 = 1, 0 + 1 = 1 + 0 = 1,

0 1 = 1 0 = 0 0 = 0, 1 1 = 1, 0 = 1, 1 = 0

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NB. Notice that 1 + 1 = 1 NOT 1 + 1 = 0. This all makes sense if wethink of 1 as TRUE, 0 as FALSE, + as or, as and, and as not.

In this course we are studying Boolean algebras in order to learn someuseful algebraic techniques for manipulating propositional formulae or settheoretic expressions. Its often easier and quicker to manipulate proposi-tional formulae or expressions involving sets algebraically than it is to relyon truth tables or Venn diagrams.

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8.3 Principle of duality for Boolean algebras

Given any arithmetic expression in a Boolean algebra, the dual of thatexpression is obtained by replacing any by a +, any + by a , and 0 bya 1 and any 1 by a 0.

The Principle of duality for Boolean algebras says that the dual of anyrule holding in a Boolean algebra must also hold. The Principle of dualityis basically a consequence of the fact that the Boolean algebra axiomsoccur in dual pairs.

We have already observed the principle of duality holding in the Booleanalgebra of the set of subsets of a set. One consequence of the principleof duality is that from any Boolean algebra S (as above) a second oneS can be defined, whose multiplication is +, addition is , unity is 0 andzero is 1.

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8.4 Rules deducible from the axioms

The following rules hold in every Boolean algebra. They are straightfor-ward to deduce from the axioms.

(a) Uniqueness of complements: that is

x + y = 1, x y = 0 y = x

(b) (x) = x.(c) x + 1 = 1 and x 0 = 0.(d) x x = x and x + x = x (idempotent laws)(e) (x + y) = x y and (x y) = x + y (de Morgans laws)

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8.5 Verifying the extra rules

The extra rules can be deduced from the axioms as follows:-

(a) Given x + y = 1, x y = 0 we havey = 1 y = (x + x) y = x y + x y = 0 + x y = x x + x y

= x (x + y) = x 1 = x

(b) We observe that x + x = x + x = 1 and x x = x x = 0 andthen apply (a) to get x = (x).(c) First we have

x + 1 = (x + 1) 1 = (x + 1) (x + x) = x + (1 x) = x + x = 1(the third equality follows from the distributive law).

The rule x 0 = 0 follows from x + 1 = 1 by duality.191

(d) We see first that

(x x) + x = (x + x) (x + x) = 1 1 = 1(applying distributivity to get the first equality) and then that

(x x) x = x (x x) = x 0 = 0So x x is the complement of x, and hence is equal to (x) = x. Thesecond idempotent law follows from the first by duality.

(e). Well verify the first of de Morgans two laws, that

(x y) = (x + y)This follows from

(x y) + (x + y) = ((x y) + x) + y = ((x + x) (y + x)) + y distrib.= (1 (y + x)) + y = (y + x) + y = (x + y) + y= x + (y + y) = x + 1 = 1, using (b)

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together with

(x y) (x + y) = x (y (x + y))= x ((y x) + (y y)) by distributivity= x ((y x) + 0) = x (y x) = x (x y)= (x x) y = 0 y = 0

The second law follows from the first by duality.

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8.6 Boolean expressions

The Boolean algebras we are interested in are in general finite. A finiteBoolean algebra must have 2n elements, for some n N , and any twoBoolean algebras of the same size are basically the same (except that theirelements may be named differently).

So from now on we shall focus on one particular Boolean algebra, theBoolean algebra of all Boolean expressions in n variables, But anythingwe prove about this Boolean algebra can be translated into a statementabout any of the other examples weve seen.

A Boolean expression in variables a, b, c, d, . . . is any expression which canbe formed by adding together strings of those symbols together with thesymbols a, b, c, . . ., and applying the axioms of Boolean algebra. Multi-plication is written as juxtaposition, addition as +, 1 is the unity and 0 the

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zero element, and x the complement of x (for any variable, or indeed anyexpression, x). This is equivalent to the Boolean algebra of propositionalformulae in a, b, c, d, on the understanding that multiplication representsand, addition or, 1 true, 0 false, and x the negation of x, and it isoften helpful to remember that.

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8.7 Tidying up Boolean expressions

Using the axioms we can tidy up any element of the Boolean algebra untilit is a sum of products of distinct variables from X together with its set ofcomplements, where no variable and its complement both appear in thesame product.

Examples. Let X = {a, b, c},(a + b)(b + c)(c + a) = (a + b)(bc + ba + cc + ca)

= (a + b)(bc + ab + ac)= abc + aab + aac + bbc + bab + bac= abc + bac = abc + abc

(a + b + c)(a + b + c) = a(a + b + c) + b(a + b + c) + c(a + b + c)= aa + ab + ac + ba + bb + bc + ca + cb + cc

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= ab + ac + ab + bc + ac + bc= ab + ab + ac + ac + bc + bc

The ordering of the terms, and within the terms, in the final expression,has been chosen for aesthetic reasons only!

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8.8 Disjunctive normal form

We can do something else with the second example above.

Since ab = ab1 = ab(c + c) = abc + abc etc., we can rewrite thefinal expression as

abc+abc+abc+abc+abc+abc+abc+abc+abc+abc+abc+abcwhich then simplifies (by reordering, and then using the idempotent lawto exclude duplicates of any term) as

abc + abc + abc + abc + abc + abcThis expression is of course more complicated than the final expressionabove. What makes it interesting is that it is a sum of terms each ofwhich is a product involving every element of X or its complement exactlyonce.

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The end result of the first example is already a sum of products of thistype.

The products like this are called the atoms (or sometimes the minterms) ofthe Boolean algebra. Any element of the Boolean algebra can be written asa sum of distinct atoms in one way only. This expression for the element isknown as its disjunctive normal form (d.n.f.) (or minterm normal form).Its sometimes useful to be able to put an element into this form, e.g. itallows us to compare elements, in particular to test if two are equal.

More examples.

Let X = {a, b, c, d} To put ab into disjunctive normal form, we writeab = ab(c + c)(d + d) = abcd + abcd + abcd + abcd

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To put ab + ac into disjunctive normal form we writeab + ac = ab(c + c)(d + d) + a(b + b)c(d + d)

= abcd + abcd + abcd + abcd + abcd + abcd + abcd + abcd= abcd + abcd + abcd + abcd + abcd + abcd

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8.9 Relating disjunctive normal form and truth tables

There is an exact correspondence between disjunctive normal forms andtruth tables. Each minterm appearing in the d.n.f of a Boolean expressioncorresponds to a row of the truth table for the expression for which thevalue of the expression is non-zero. The corresponding row is that row atwhich a variable x takes the value 1 if x itself appears in the minterm,but the value 0 if x appears in the minterm.Hence we can deduce the truth table of an expression from its d.n.f., orthe d.n.f. from the truth table.

e.g. consider the two examples in variables {a, b, c, d} aboves, for whichwe already computed the d.n.f. algebraically..

The truth table for ab has non-zero values on the rows at which (a, b, c, d)

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takes the values (1, 1, 1, 1), (1, 1, 1, 0), (1, 1, 0, 1), (1, 1, 0, 0).

The truth table for ab + ac has non-zero values on the 6 rows where(a, b, c, d) takes the values

(1, 0, 1, 1), (1, 0, 1, 0), (1, 0, 0, 1), (1, 0, 0, 0), (1, 1, 0, 1), (1, 1, 0, 0).

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