CSci 2011 Discrete Mathematics Lecture 8 CSci 2011
CSci 2011 Discrete
Mathematics
Lecture 8
CSci 2011
CSci 2011
Admin Due dates and quiz
Groupwork 4 is due on Oct 5th. Homework 3 is due on Oct 14th. Quiz 2: Oct 7rd.
1 page cheat sheet is allowed.
E-mail: [email protected] Put [2011] in front.
Me and Aziz will be out of town next week. No office hour for Yongdae Aziz’s office hour will be replaced by someone.
Check class web site Read syllabus, Use forum.
Study Guides
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RecapPropositional operation summary
Check translationDefinition
Tautology, Contradiction, logical equicalence
not not and or conditional Bi-conditional
p q p q pq pq pq pq
T T F F T T T T
T F F T F T F F
F T T F F T T F
F F T T F F T T
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Recap
p T pp F p
Identity Laws(p q) r p (q r)(p q) r p (q r)
Associative laws
p T Tp F F
Domination Lawp (q r) (p q) (p r)p (q r) (p q) (p r)
Distributive laws
p p pp p p
Idempotent Laws
(p q) p q (p q) p q
De Morgan’s laws
( p) p Double negation law
p (p q) pp (p q) p
Absorption laws
p q q pp q q p
Commutative Laws
p p Tp p F
Negation lows
pq pq Definition of Implication p q (p q) (q p) Definition of
Biconditional
Recap Quantifiers
Universal quantifier: x P(x) Negating quantifiers
¬x P(x) = x ¬P(x) ¬x P(x) = x ¬P(x) xy P(x, y)
Nested quantifiers xy P(x, y): “For all x, there exists a y such that P(x,y)” xy P(x,y): There exists an x such that for all y P(x,y) is true” ¬x P(x) = x ¬P(x), ¬x P(x) = x ¬P(x)
Proof techniquesDirect proof Indirect Proof
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Recap
Modus ponens
p p q q
Modus tollens
q p q p
Hypothetical syllogism
p q q r p r
Disjunctive syllogism
p q p q
Addition p p q
Simplification p q p
Conjunction
p q p q
Resolution
p q p r q r
Recap p→q
Direct Proof: Assume p is true. Show that q is also true. Indirect Proof: Assume ¬q is true. Show that p is true.
Proof by contradiction Proving p: Assume p is not true. Find a contradiction. Proving p→q
¬(p→q) (p q) p ¬q Assume p is tue and q is not true. Find a contradiction.
Proof by Cases: [(p1p2…pn)q] [(p1q)(p2q)…(pnq)]
If and only if proof: pq (p→q)(q→p) Existence Proof
Constructive vs. Non-constructive Proof
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Uniqueness proofsA theorem may state that only one such
value exists
To prove this, you need to show:Existence: that such a value does indeed exist
Either via a constructive or non-constructive existence proof
Uniqueness: that there is only one such value
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Uniqueness proof example If the real number equation 5x+3=a has a solution
then it is unique
Existence We can manipulate 5x+3=a to yield x=(a-3)/5 Is this constructive or non-constructive?
Uniqueness If there are two such numbers, then they would fulfill the
following: a = 5x+3 = 5y+3 We can manipulate this to yield that x = y
Thus, the one solution is unique!
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Forward and Backward Reasoning
(x+y) / 2 > √xy for all distinct positive real number x and y.
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Counterexamples Given a universally quantified statement, find a single example
which it is not true
Note that this is DISPROVING a UNIVERSAL statement by a counterexample
x ¬R(x), where R(x) means “x has red hair” Find one person (in the domain) who has red hair
Every positive integer is the square of another integer The square root of 5 is 2.236, which is not an integer
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What’s wrong with this proof?If n2 is an even integer, then n is an even
integer.Proof) Suppose n2 is even. Then n2 = 2 k for
some integer k. Let n = 2 l for some integer l. Then n is an even integer.
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Proof methods We will discuss ten proof methods:
1. Direct proofs2. Indirect proofs3. Vacuous proofs4. Trivial proofs5. Proof by contradiction6. Proof by cases7. Proofs of equivalence8. Existence proofs9. Uniqueness proofs10. Counterexamples
ch2.1 Sets
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What is a set? A set is a unordered collection of “objects”
People in a class: {Alice, Bob, Chris } States in the US: {Alabama, Alaska, Virginia, … } Sets can contain non-related elements: {3, a, red, Virginia }
We will most often use sets of numbers All positive numbers less than or equal to 5: {1, 2, 3, 4, 5} A few selected real numbers: { 2.1, π, 0, -6.32, e }
Properties Order does not matter
{1, 2, 3, 4, 5} is equivalent to {3, 5, 2, 4, 1} Sets do not have duplicate elements
Consider the list of students in this class– It does not make sense to list somebody twice
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Specifying a Set Capital letters (A, B, S…) for sets Italic lower-case letter for elements (a, x, y…)
Easiest way: list all the elements A = {1, 2, 3, 4, 5}, Not always feasible!
May use ellipsis (…): B = {0, 1, 2, 3, …} May cause confusion. C = {3, 5, 7, …}. What’s next? If the set is all odd integers greater than 2, it is 9 If the set is all prime numbers greater than 2, it is 11
Can use set-builder notation D = {x | x is prime and x > 2} E = {x | x is odd and x > 2} The vertical bar means “such that”
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Specifying a setA set “contains” the various “members” or
“elements” that make up the set If an element a is a member of (or an element of)
a set S, we use then notation a S 4 {1, 2, 3, 4}
If not, we use the notation a S 7 {1, 2, 3, 4}
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Often used sets N = {0, 1, 2, 3, …} is the set of natural numbers Z = {…, -2, -1, 0, 1, 2, …} is the set of integers Z+ = {1, 2, 3, …} is the set of positive integers
(a.k.a whole numbers) Note that people disagree on the exact definitions of whole
numbers and natural numbers Q = {p/q | p Z, q Z, q ≠ 0} is the set of rational
numbers Any number that can be expressed as a fraction of two
integers (where the bottom one is not zero) R is the set of real numbers
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The universal set 1U is the universal set – the set of all of
elements (or the “universe”) from which given any set is drawnFor the set {-2, 0.4, 2}, U would be the real
numbersFor the set {0, 1, 2}, U could be the N, Z, Q, R
depending on the contextFor the set of the vowels of the alphabet, U would
be all the letters of the alphabet
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Venn diagrams Represents sets graphically
The box represents the universal set Circles represent the set(s)
Consider set S, which is the set of all vowels in thealphabet
The individual elements are usually not written in a Venn diagram
a e i
o u
b c d fg h jk l mn p q
r s tv w xy z
US
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Sets of setsSets can contain other sets
S = { {1}, {2}, {3} }T = { {1}, {{2}}, {{{3}}} }V = { {{1}, {{2}}}, {{{3}}}, { {1}, {{2}},
{{{3}}} } } V has only 3 elements!
Note that 1 ≠ {1} ≠ {{1}} ≠ {{{1}}}They are all different
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The Empty Set If a set has zero elements, it is called the empty (or
null) set Written using the symbol Thus, = { } VERY IMPORTANT
It can be a element of other sets { , 1, 2, 3, x } is a valid set
≠ { } The first is a set of zero elements The second is a set of 1 element
Replace by { }, and you get: { } ≠ {{ }} It’s easier to see that they are not equal that way
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Set Equality, Subsets Two sets are equal if they have the same elements
{1, 2, 3, 4, 5} = {5, 4, 3, 2, 1} {1, 2, 3, 2, 4, 3, 2, 1} = {4, 3, 2, 1} Two sets are not equal if they do not have the same
elements {1, 2, 3, 4, 5} ≠ {1, 2, 3, 4}
If all the elements of a set S are also elements of a set T, then S is a subset of T If S = {2, 4, 6}, T = {1, 2, 3, 4, 5, 6, 7}, S is a subset of T This is specified by S T meaning that x (x S x T)
For any set S, S S (S S S) For any set S, S (S S)
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If S is a subset of T, and S is not equal to T, then S is a proper subset of T Can be written as: R T and R T Let T = {0, 1, 2, 3, 4, 5} If S = {1, 2, 3}, S is not equal to T, and S is a subset of T A proper subset is written as S T Let Q = {4, 5, 6}. Q is neither a subset or T nor a proper
subset of T
The difference between “subset” and “proper subset” is like the difference between “less than or equal to” and “less than” for numbers
Proper Subsets
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Set cardinality The cardinality of a set is the number of elements in
a set, written as |A|
Examples Let R = {1, 2, 3, 4, 5}. Then |R| = 5 || = 0 Let S = {, {a}, {b}, {a, b}}. Then |S| = 4
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Power Sets Given S = {0, 1}. All the possible subsets of S?
(as it is a subset of all sets), {0}, {1}, and {0, 1} The power set of S (written as P(S)) is the set of all the
subsets of S P(S) = { , {0}, {1}, {0,1} }
Note that |S| = 2 and |P(S)| = 4
Let T = {0, 1, 2}. The P(T) = { , {0}, {1}, {2}, {0,1}, {0,2}, {1,2}, {0,1,2} }
Note that |T| = 3 and |P(T)| = 8
P() = { } Note that || = 0 and |P()| = 1
If a set has n elements, then the power set will have 2n elements
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Tuples In 2-dimensional space, it is a (x, y) pair of numbers
to specify a location In 3-dimensional (1,2,3) is not the same as (3,2,1) –
space, it is a (x, y, z) triple of numbers In n-dimensional space, it is a
n-tuple of numbers Two-dimensional space uses
pairs, or 2-tuples Three-dimensional space uses
triples, or 3-tuples Note that these tuples are
ordered, unlike sets the x value has to come first +x
+y
(2,3)
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Cartesian products A Cartesian product is a set of all ordered 2-tuples
where each “part” is from a given set Denoted by A x B, and uses parenthesis (not curly brackets) For example, 2-D Cartesian coordinates are the set of all
ordered pairs Z x Z Recall Z is the set of all integers This is all the possible coordinates in 2-D space
Example: Given A = { a, b } and B = { 0, 1 }, what is their Cartiesian product?
C = A x B = { (a,0), (a,1), (b,0), (b,1) }
Formal definition of a Cartesian product: A x B = { (a,b) | a A and b B }
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Cartesian Products 2 All the possible grades in this class will be a
Cartesian product of the set S of all the students in this class and the set G of all possible grades Let S = { Alice, Bob, Chris } and G = { A, B, C } D = { (Alice, A), (Alice, B), (Alice, C), (Bob, A), (Bob, B),
(Bob, C), (Chris, A), (Chris, B), (Chris, C) } The final grades will be a subset of this: { (Alice, C), (Bob,
B), (Chris, A) } Such a subset of a Cartesian product is called a relation (more
on this later in the course)
ch2.2 Set Operations
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Set operations: Union Formal definition for the union of two sets:
A U B = { x | x A or x B }
Further examples {1, 2, 3} U {3, 4, 5} = {1, 2, 3, 4, 5} {a, b} U {3, 4} = {a, b, 3, 4} {1, 2} U = {1, 2}
Properties of the union operation A U = A Identity law A U U = U Domination law A U A = A Idempotent law A U B = B U A Commutative law A U (B U C) = (A U B) U C Associative law
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Set operations: Intersection Formal definition for the intersection of two sets: A ∩
B = { x | x A and x B }
Examples {1, 2, 3} ∩ {3, 4, 5} = {3} {a, b} ∩ {3, 4} = {1, 2} ∩ =
Properties of the intersection operation A ∩ U = A Identity law A ∩ = Domination law A ∩ A = A Idempotent law A ∩ B = B ∩ A Commutative law A ∩ (B ∩ C) = (A ∩ B) ∩ C Associative law
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Disjoint setsFormal definition for disjoint sets: two sets
are disjoint if their intersection is the empty set
Further examples{1, 2, 3} and {3, 4, 5} are not disjoint{a, b} and {3, 4} are disjoint{1, 2} and are disjoint
Their intersection is the empty set and are disjoint!
Their intersection is the empty set
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Formal definition for the difference of two sets:A - B = { x | x A and x B }
Further examples{1, 2, 3} - {3, 4, 5} = {1, 2}{a, b} - {3, 4} = {a, b}{1, 2} - = {1, 2}
The difference of any set S with the empty set will be the set S
Set operations: Difference
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Complement setsFormal definition for the complement of a
set: A = { x | x A } = Ac
Or U – A, where U is the universal set
Further examples (assuming U = Z){1, 2, 3}c = { …, -2, -1, 0, 4, 5, 6, … }{a, b}c = Z
Properties of complement sets (Ac)c = A Complementation lawA U Ac = U Complement lawA ∩ Ac = Complement law
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Set identities
A = AAU = A
Identity LawAU = UA =
Domination law
AA = AAA = A
Idempotent Law
(Ac)c = AComplement
Law
AB = BAAB = BA
Commutative Law
(AB)c = AcBc
(AB)c = AcBc
De Morgan’s Law
A(BC) = (AB)CA(BC)
= (AB)C
Associative Law
A(BC) = (AB)(AC)A(BC) =
(AB)(AC)
Distributive Law
A(AB) = AA(AB) = A
Absorption Law
A Ac = UA Ac =
Complement Law
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How to prove a set identityFor example: A∩B=B-(B-A)Four methods:
Use the basic set identitiesUse membership tablesProve each set is a subset of each otherUse set builder notation and logical equivalences
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What we are going to prove…A∩B=B-(B-A)
A B
A∩B B-AB-(B-A)
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Proof by Set IdentitiesA B = A - (A - B)Proof) A - (A - B) = A - (A Bc)
= A (A Bc)c
= A (Ac B) = (A Ac) (A B) = (A B) = A B
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Showing each is a subset of the others
(A B)c = Ac Bc
Proof) Want to prove that (A B)c Ac Bc and (A B)c Ac Bc
x (A B)c
x (A B) (x A B) (x A x B) (x A) (x B) x A x B x Ac x Bc
x Ac Bc
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Examples Let A, B, and C be sets. Show that:a) (AUB) (AUBUC)b) (A∩B∩C) (A∩B)c) (A-B)-C A-Cd) (A-C) ∩ (C-B) =