University of Aberdeen, Computing Science CS2013 Mathematics for Computing Science Adam Wyner Slides adapted from Michael P. Frank ’ s course based on.

University of Aberdeen, Computing Science

CS2013Mathematics for Computing

ScienceAdam Wyner

Slides adapted fromMichael P. Frank’s course based on the textDiscrete Mathematics & Its Applications

(5th Edition)by Kenneth H. Rosen

Propositional Logic

Rosen 5th ed., §§1.1-1.2 (but much extended)~85 slides, ~2 lectures

Frank / van Deemter / Wyner 3

Foundations of Logic

Mathematical Logic is a tool for working with compound statements. It includes:

• A formal language for expressing them.• A methodology for reasoning about their

truth or falsity.• It is the foundation for expressing formal

proofs in all branches of mathematics.

Fall 2013


Two Logical Systems:

1. Propositional logic

2. Predicate logic (extends 1. )

Many other logical `calculi` exist, but they tend to resemble these two

Fall 2013


Propositional Logic (§1.1)

Propositional Logic is the logic of compound statements built from simpler statements using so-called Boolean connectives.

Some applications in computer science:• Design of digital electronic circuits.• Expressing conditions in programs.• Queries to databases & search engines.

George Boole(1815-1864)

Chrysippus of Soli(ca. 281 B.C. – 205 B.C.)Fall 2013


Propositions in natural language

In propositional logic, a proposition is simply:• a statement (i.e., a declarative sentence)

–with some definite meaning• having a truth value that’s either true (T) or false

(F). Only values statements can have.–Never both, or somewhere “in between”.

However, you might not know the truth value

Fall 2013


Examples of NL Propositions

• “It is raining.” (In a given situation.)• “Beijing is the capital of China, and 1 + 2 = 2”

But, the following are NOT propositions:• “Who’s there?” (interrogative: no truth value)• “x := x+1” (imperative: no truth value)• “1 + 2” (term: no truth value)

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Propositions in Propositional Logic

• Atoms: p, q, r, …(Corresponds with simple English sentences, e.g.‘I had salad for lunch’)

• Complex propositions : built up from atoms using operators: pq (Corresponds with compound English sentences, e.g., “I had salad for lunch and I had steak for dinner.”)

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Defining Propositions

• Logic defines notions of atomic and complex propositions and what complex propositions “mean”.

• We explain by example, giving precise definitions.

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An operator or connective combines with n operand expressions into a larger expression.

• Unary operators take 1 operand;• Binary operators take 2 operands.• Propositional or Boolean operators operate

on propositions instead of on numbers (+,-).

Operators / Connectives

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Common Boolean Operators

Formal Name Nickname Arity Symbol

Negation operator NOT Unary ¬

Conjunction operator AND Binary

Disjunction operator OR Binary

Exclusive-OR operator XOR Binary

Implication operator IMPLIES Binary

Biconditional operator IFF Binary ↔

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The Negation Operator

The unary negation operator “¬” (NOT) combines with one prop, transforming the prop into its negation.

E.g. If p = “I have brown hair.”

then ¬p = “I do not have brown hair.”

The truth table for NOT: p p T F F T

T :≡ True; F :≡ False“:≡” means “is defined as”

Operandcolumn

ResultcolumnFall 2013


Truth-functionality

• Truth table expresses truth/falsity of ¬p in terms of truth/falsity of p

• This not possible for the operator ‘tomorrow’, or `probably’: – ‘Tomorrow p’ is true iff p is ….’??– ‘Probably p’ is true iff p is ….’??

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Truth-functionality

• Truth table expresses truth/falsity of ¬p in terms of truth/falsity of p.

• Each horizontal line of the table expresses some alternative context.

• Truth-functional operator: an operator that is a function from the truth values of the component expressions to a truth value.

• NOT is truth functional. Yesterday is not.• Propositional logic is only about truth-functional

operators.• We can compute the values of the complex expressions.

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Comment on Truth

• Not concerned with the "meaning" of a proposition other than whether it is true or false. Not about "how" we know it is true or false, but supposing it is, what else do we know. Abstraction.

• The "truth" of a proposition determined "by inspection" – The book is on the table. The "real" world.

• The "truth" determined by "stipulation"- suppose The book is on the table is true. Not the "real" world.

• Mostly we are stipulate truth of a proposition. The truth value of P is true/false.

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The Conjunction Operator

The binary conjunction operator “” (AND) combines two propositions to form their logical conjunction.

E.g. If p=“I will have salad for lunch.” and q=“I will have steak for dinner.”, then pq=“I will have salad for lunch and I will have steak for dinner.”

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• Note that aconjunctionp1 p2 … pn

of n propositionswill have 2n rowsin its truth table.

• Also: ¬ and operations together can express any Boolean truth table!

Conjunction Truth Table

Operand columns

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more later


The Disjunction Operator

The binary disjunction operator “” (OR) combines two propositions to form their logical disjunction.

p=“My car has a bad engine.”

q=“My car has a bad carburator.”

pq=“Either my car has a bad engine, or my car has a bad carburetor.”

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• Note that pq meansthat p is true, or q istrue, or both are true!

• So, this operation isalso called inclusive or,because it includes thepossibility that both p and q are true.

• “¬” and “” together are also universal.

Disjunction Truth Table

p q p qF F FF T TT F TT T T

Notedifferencefrom AND

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Nested Propositional Expressions

• Use parentheses to group sub-expressions:“I just saw my old friend, and either he’s grown or I’ve shrunk.” = f (g s)(f g) s would mean something different

f g s would be ambiguous

• By convention, “¬” takes precedence over both “” and “”.¬s f means (¬s) f rather than ¬ (s f)

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Logic as shorthand for NL

Let p=“It rained last night”, q=“The sprinklers came on last night,” r=“The lawn was wet this morning.”

¬p =

r ¬p =

¬ r p q =

It didn't rain last night.

The lawn was wet this morning, andit didn’t rain last night.Either the lawn wasn't wet this morning, or it rained last night, or the sprinklers came on last night.

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Some important ideas:

• Distinguishing between different kinds of formulas

• Seeing that some formulas that look different may express the same information

• First: different kinds of formulas

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Tautologies

A tautology is a compound proposition that is true no matter what the truth values of its atomic propositions are!

Ex. p p [What is its truth table?]

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Tautologies

• When every row of the truth table gives T.

• Example: p p T T FT F T TF

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Contradictions

A contradiction is a compound proposition that is false no matter what! Ex. p p [Truth table?]

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Contradictions

• When every row of the truth table gives F

• Example: p p T F FT F F TF

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Contingencies

All other props. are contingencies:

Some rows give T, others give F

Now: formulas that have the same meaning

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Propositional Equivalence

Two syntactically (i.e., textually) different compound propositions may be semantically identical (i.e., have the same meaning). Here semantically identical means just that they have the same truth table for input truth values of the propositions.

We call them logically equivalent.

Notation: … …

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Logical Equivalence

Compound proposition p is logically equivalent to compound proposition q, written pq, IFF p and q contain the same truth values in all rows of their truth tables

We will also say: they express the same truth function (= the same function from values for atoms to values for the whole formula).

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Ex. Prove that pq (p q).

p q pp qq pp qq pp qq ((pp qq))F FF TT FT T

Proving Equivalencevia Truth Tables

FT

TT

T

T

T

TTT

FF

F

F

FFF

F

TT

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Shows that OR is equivalent to a combination of NOT and AND.


Before introducing more connectives

• … let us step back and ask a few questions about truth tables

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1. What does each line of the table "mean"?

2. Consider a conjunction p1 p2 p3

How many rows are there in its truth table?

3. Consider a conjunctionp1 p2 … pn of n propositions.How many rows are there in its truth table?

4. Explain why ¬ and together are sufficient to express any Boolean truth table

Questions for you to think about

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1. Consider a conjunction p1 p2 p3

How many rows are there in its truth table? 8 p1 p2 p3 1 1 1 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0


Fall 2013

Two truth values (0,1) and three propositions:

23 = 8.


2. Consider p1 p2 … pn How many rows are there in its truth table?

2*2*2* … *2 (n factors)Hence 2n (This grows exponentially!)


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3. Explain why ¬ and together are sufficient to express any other complex expression in propositional logic.


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3. Explain why ¬ and together are sufficient to express any other complex expression in propositional logic.

• Obviously, if we add new connectives(like ) we can write new formulas.

• CLAIM: these formulas would always be equivalent with ones that only use ¬ and (This is what we need to prove).


Fall 2013


• Saying this in a different way: if we add new connectives, we can write new formulas, but these formulas will always only express truth functions that can already be expressed by formulas that only use ¬ and .

• That is, they will be equivalent.• Example of writing a disjunction in another

form (equivalence shown before):

p q ¬(¬p ¬q)

Relating AND and OR

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Mystery Operator

PQR Formula (containing P,Q,R)1 1 1 01 1 0 11 0 1 11 0 0 00 1 1 00 1 0 00 0 1 00 0 0 0 Does there exist such a Formula?

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Suppose, given the truth values of P, Q, and R, we construct a Formula with the given resulting truth value. This is our 'mystery' operator . Can it be written equivalently with NOT and AND.


3. Explain why ¬ and together are sufficient to express any Boolean truth table

• Suppose precisely two rows give T.For example, the rows where

– P=T, Q=T, R=F. This is P Q ¬R– P=T, Q=F, R=T. This is P ¬Q R

T-values in Conjunction

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Suppose precisely two rows give T.For example, the rows where

– P=T, Q=T, R=F. This is P Q ¬R– P=T, Q=F, R=T. This is P ¬Q R

• We’ve proven our claim if we can express the disjunction of these two rows: (P Q ¬R) (P ¬Q R)

Table as a disjunction of T-rows

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• We’ve arrived if we can express their disjunction: (P Q ¬R) (P ¬Q R)

• But we’ve seen that disjunction can be expressed using and ¬: A B ¬(¬A ¬B)

• So: (P Q ¬R) (P ¬Q R) ¬(¬ (P Q ¬R) (P ¬Q R))

• We’ve only used and .

Disjoining rows of the table

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(P Q ¬R) (P ¬Q R) ¬(¬ (P Q ¬R) (P ¬Q R)) 1 1 0 01 0 1 01 0 1 0 1 1 1 0 01 1 1 1 01 0 1 1 1 1 10 1 1 01 0 0 1 0 1 1 1 10 0 1 1 01 0 0 1 0 0 01 1 1 10 1 1 1 1 1 0 0 01 0 0 1 10 1 1 1 0 0 10 0 1 10 0 0 0 1 1 0 0 10 1 1 1 10 0 0 0 1 0 01 0 0 01 0 1 0 1 0 1 0 01 1 1 0 01 0 1 0 1 0 10 0 0 01 0 0 0 1 0 1 0 10 1 1 0 01 0 0 0 0 0 01 0 0 10 0 1 0 1 0 0 0 01 1 1 0 10 0 1 0 0 0 10 0 0 10 0 0 0 1 0 0 0 10 1 1 0 10 0 0

Check

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Notice that we only state where the new mystery connective is true as it is false elsewhere.


About this proof …

• We’ve made our task a bit easier, assuming that there were only 2 rows resulting in T

• But the case with 1 or 3 or 4 or …. rows is analogous (and there are always only finitely many rows.)

• So, the proof can be made precise

Fall 2013


Having proved this …

• We can express every possible truth-functional operator in propositional logic in terms of AND and NOT

• This is sometimes called functional completeness. Also universality.

• Reduce other operators to other more basic operators.• Very useful in computing to reduce complexity of formulas

(Normal forms)

Fall 2013


Let’s introduce some additional connectives

• A variant of disjunction• The conditional• The biconditional

Fall 2013


The Exclusive Or Operator

The binary exclusive-or operator “” (XOR) combines two propositions to form their logical “exclusive or”.

p = “I will earn an A in this course,”

q = “I will drop this course,”

p q = “I will either earn an A in this course, or I will drop it (but not both!)”

Fall 2013


• Note that pq meansthat p is true, or q istrue, but not both!

• This operation iscalled exclusive or,because it excludes thepossibility that both p and q are true.

• “¬” and “” together are not universal.

Exclusive-Or Truth Table

p q pqF F FF T TT F TT T F Note

differencefrom OR.

Fall 2013


Note that English “or” can be ambiguous regarding the “both” case!

Need context to disambiguate the meaning!For this class, assume “or” means inclusive.

Natural Language is Ambiguous

p q p "or" qF F FF T TT F TT T ?

Fall 2013


Test your understanding of the two types of disjunction

1. Suppose p q is true.Does it follow that pq is true?

2. Suppose pq is true.Does it follow that p q is true?

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Test your understanding of the two types of disjunction

1. Suppose p q is true.Does it follow that pq is true?No: consider p TRUE, q TRUE

2. Suppose pq is true. Does it follow that p q is true? Yes. Check each of the two assignments that make pq true: a) p TRUE, q FALSE (makes p q true) b) p FALSE, q TRUE (makes p q true)

Fall 2013


The Implication Operator

The implication p q states that p implies q.

I.e., If p is true, then q is true; but if p is not true, then q could be either true or false.

E.g., let p = “You study hard.” q = “You will get a good grade.”

p q = “If you study hard, then you will get a good grade.”

antecedent consequent

Fall 2013


Implication Truth Table

• p q is false only when(p is true but q is not true)

• p q does not saythat p causes q!

• p q does not requirethat p or q are true!

• E.g. “(1=0) pigs can fly” is TRUE!

p q p q F F T F T T T F F T T T

The onlyFalsecase!

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• Suppose you know that q is T. What do you know aboutpq ?


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• Suppose you know that q is T. What do you know aboutpq ?

• The conditional must be T


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• Suppose you knowthat p is F. Whatdo you know aboutpq ?

• The conditionalmust be T


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• Suppose you knowthat p is T.

• What do you know aboutpq ? T or F.

• What do you know about

q? T or F.


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Implications between real sentencs

• “If this lecture ever ends, then the sun has risen this morning.” True or False?

• “If Tuesday is a day of the week, then I am a penguin.” True or False?

• “If 1+1=6, then Bush is president.” True or False?

• “If the moon is made of green cheese, then 1+1=7.” True or False?

Fall 2013


Why does this seem wrong?

• Recall “If [you study hard] then [you’ll get a good grade]”

• In normal English, this asserts a causal connection between the two propositions. The connective does not capture this connection.

Fall 2013


English Phrases Meaning p q

• “p implies q”• “if p, then q”• “if p, q”• “when p, q”• “whenever p, q”• “q if p”• “q when p”• “q whenever p”

• “p only if q”• “p is sufficient for q”• “q is necessary for p”• “q follows from p”• “q is implied by p”

Fall 2013

Tricky: "only if" is p q. "p only if q" says that p cannot be true when q is not true. The statement "p only if q" is false where p is true, but q is false. Where p is true, then so is q.


But we’re not studying English ..

• Probably no two of these expressions have exactly the same meaning in English

• For example, ‘I’ll go to the party if Mary goes’can be interpreted as implying‘I’ll only go to the party if Mary goes’turning the sentence into a biconditional:I go IFF Mary goes

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Biconditional Truth Table

• p q means that p and qhave the same truth value.

• Note this truth table is theexact opposite of ’s!Thus, p q means ¬(p q)

• p q does not implythat p and q are true, or that either of them causes the other.

p q p qF F TF T FT F FT T T

Fall 2013


Consider ...

The truth of p q, where

1. p= Scotland is in the UKq= 2+2 =4

2. p= Scotland is not in the UKq= 2+2 =5

3. p= Scotland is in the UKq= Wales is not in the UK

Fall 2013


Consider ...


1. p= Scotland is in the UKq= 2+2 =4 TRUE

2. p= Scotland is not in the UKq= 2+2 =5 TRUE

3. p= Scotland is in the UKq= Wales is not in the UK FALSE

Fall 2013


The difference between and

A B says that A and B happen to have the same truth value. They could be atomic.

A B says that no assignment of truth values to A and B can make A B false

So, A B can only hold between well-chosen compound A and B.

A B says “A B is a tautology”

Fall 2013


Consider ...


1. p= Scotland is in the UKq= 2+2 =4

2. p= Scotland is not in the UKq= Wales is not in the UK

3. p= Scotland is in UK Wales is in UKq= Wales is in UK Scotland is in UK

Fall 2013


Consider ...

The truth of pq, where1. p= Scotland is in the UK

q= 2+2 =4 FALSE2. p= Scotland is not in the UK

q= Wales is not in the UK FALSE3. p= Scotland is in UK Wales is in UK

q= Wales is in UK Scotland is in UK TRUE

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The language of propositional logic defined more properly (i.e., as a formal language)

• Atoms: p1, p2, p3, ..• Formulas:

– All atoms are formulas– For all , if is a formula then ¬ is a formula– For all and , if and are formulas then the

following are formulas: ( ), ( ), ( ) (etc.)

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• Which of these are formulas, according to this strict definition?

• p1 ¬ p2 • (p1 ¬ p2) • ¬ ¬ ¬(p9 p8)• (p1 p2 p3)• (p1 (p2 p3))

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• Which of these are formulas, according to this strict definition?

• p1 ¬ p2 No• (p1 ¬ p2) Yes• ¬ ¬ ¬(p9 p8) Yes• (p1 p2 p3) No• (p1 (p2 p3)) Yes

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Simplifying conventions

• Convention 1: outermost brackets may be omitted,:p1 ¬ p2, ¬ ¬ ¬(p9 p8), p1 (p2 p3)

• Convention 2: associativity allows us to omit even more brackets, e.g.:p1 p2 p3, p1 p2 p3

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• Which of these are formulas, when using these two conventions?

• p1 ¬ p2 Yes• (p1 ¬ p2) Yes• ¬ ¬ ¬(p9 p8) Yes• (p1 p2 p3) No• (p1 (p2 p3)) Yes

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Contrapositive

Some terminology, for an implication p q:• Its converse is: q p.• Its contrapositive: ¬q ¬ p.

• Which of these two has/have the same meaning (express same truth function) as p q? Prove it.

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Contrapositive

Proving the equivalence of p q and its contrapositive, using truth tables:

p q q p p q q p F F T T T T F T F T T T T F T F F F T T F F T T

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Boolean Operations Summary

• We have seen 1 unary operator (out of the 4 possible ones) and 5 binary operators:

p q p p q p q pq p q pqF F T F F F T TF T T F T T T FT F F F T T F FT T F T T F T T

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For you to think about:

1. Can you think of yet another 2-place connective?How many possible connectives do there exist?

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For you to think about:

1. How many possible connectives do there exist?

p connective q T ? T T ? F F ? T F ? FEach question mark can be T or F, hence2*2*2*2=16 connectives

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Example of another connective

p connective q compare: p and q

T F T TT T F FF T T FF T F F

Names: NAND, Sheffer stroke

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Some Alternative Notations

Name: notandorxorimplies iffPropositional logic: Boolean algebra: ppq+C/C++/Java (wordwise):!&&||!= ==C/C++/Java (bitwise): ~&|^Logic gates:

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To this point

You have learned about:

• Propositional logic operators’– Symbolic notations.– English equivalents– Truth tables.– Logical equivalence

• Next:– More about logical equivalences.– How to prove them.

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Tautologies revisited

• We’ve introduced the notion of a tautology using the example p p

• Now, you know more operators, so can formulate many more tautologies, e.g.,the following are tautologies:(pq) (p q)(pq) (¬q ¬ p), and so on

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Equivalence Laws

• Similar to arithmetic identities in algebra• Patterns that can be used to match (part of)

another proposition• Abbreviation: T stands for an arbitrary

tautology; F an arbitrary contradiction

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Equivalence Laws - Examples

• Identity: pT p pF p• Domination: pT T pF F• Idempotence: pp p pp p• Double negation: p p• Commutativity: pq qp pq qp• Associativity: (pq)r p(qr)

(pq)r p(qr)

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More Equivalence Laws

• Distributive: p(qr) (pq)(pr) p(qr) (pq)(pr)

• De Morgan’s:(pq) p q

(pq) p q • Trivial tautology/contradiction:

p p T p p FAugustus

De Morgan(1806-1871)

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Defining Operators via Equivalences

Some equivalences can be thought of as definitions of one operator in terms of others:

• Exclusive or: pq (pq)(pq) pq (pq)(qp)

• Implies: pq p q• Biconditional: pq (pq) (qp)

pq (pq)

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How you may use equivalence laws: Example (1)

• Use equivalences to prove that (r s) s r.

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(r s) (De Morgan)

r s (Commutativity)

s r (2x Double Negation)

s r.

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• Use equivalences to prove that (p q) (p r) p q r.

• (This would be much easier using truth tables!)

(p q) (p r) [Expand “definition” of ] (p q) (p r) [Expand “definition” of ]

(p q) ((p r) (p r)) [DeMorgan]

(p q) ((p r) (p r)) cont.

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Underline just focuses interest.


Long example Continued...

(p q) ((p r) (p r)) [ commutes]

(q p) ((p r) (p r)) [ is associative]

q (p ((p r) (p r))) [distrib. over ]Û q (((p (p r)) (p (p r))) [assoc.]Û q (((p p) r) (p (p r))) [taut.] Û q ((T r) (p (p r))) [domination] Û q (T (p (p r))) [identity] q (p (p r)) cont.

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End of Long Example

q (p (p r)) [DeMorgan] Û q (p (p r)) [Assoc.] Û q ((p p) r) [Idempotent] Û q (p r) [Associativity] Û (q p) r [Commutativity]

p q r

Q.E.D. (quod erat demonstrandum)

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Summary

• In practice, Propositional Logic equivalences are seldom strung together into long proofs: using truth tables is usually much easier.

• We shall now turn to a more complex logic, where nothing like truth tables is available.

Fall 2013

University of Aberdeen, Computing Science CS2013 Mathematics for Computing Science Adam Wyner Slides adapted from Michael P. Frank ’ s course based on.

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