1. CSci 2011 Discrete Mathematics Lecture 8 CSci 2011

2. Admin ^Due dates and quiz  Groupwork 4 is due on Oct 5 th.  Homework 3 is due on Oct 14 th.  Quiz 2: Oct 7 rd.  1 page cheat sheet is allowed. ^E-mail: csci2011-help@cselabs.umn.edu csci2011-help@cselabs.umn.edu  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

3. CSci 2011 Recap ^Propositional operation summary ^Check translation ^Definition  Tautology, Contradiction, logical equicalence not andorconditionalBi-conditional pq ppqqpqpqpqpqpqpqpqpq TTFFTTTT TFFTFTFF FTTFFTTF FFTTFFTT

4. CSci 2011 Recap p  T  p p  F  p Identity Laws (p  q)  r  p  (q  r) (p  q)  r  p  (q  r) Associative laws p  T  T p  F  F Domination Law p  (q  r)  (p  q)  (p  r) p  (q  r)  (p  q)  (p  r) Distributive laws p  p  p p  p  p Idempotent Laws  (p  q)   p   q  (p  q)   p   q De Morgan’s laws ( p)  p Double negation law p  (p  q)  p p  (p  q)  p Absorption laws p  q  q  p p  q  q  p Commutative Laws p   p  T p   p  F Negation lows pq  pq Definition of Implication p  q  (p  q)  (q  p) Definition of Biconditional

5. Recap ^Quantifiers  Universal quantifier: x P(x)  Negating quantifiers  ¬x P(x) = x ¬P(x)  ¬x P(x) = x ¬P(x) xy P(x, y)  Nested quantifiers  xy P(x, y): “For all x, there exists a y such that P(x,y)”  xy 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 techniques  Direct proof  Indirect Proof CSci 2011

6. 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

7. 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: [(p 1 p 2 …p n )q]  [(p 1 q)(p 2 q)…(p n q)]  If and only if proof: pq (p → q)(q → p) ^Existence Proof  Constructive vs. Non-constructive Proof CSci 2011

8. Uniqueness proofs ^A 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

9. CSci 2011 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!

10. CSci 2011 Forward and Backward Reasoning ^(x+y) / 2 > √xy for all distinct positive real number x and y.

11. CSci 2011 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

12. CSci 2011 What’s wrong with this proof? ^If n 2 is an even integer, then n is an even integer. Proof) Suppose n 2 is even. Then n 2 = 2 k for some integer k. Let n = 2 l for some integer l. Then n is an even integer.

13. CSci 2011 Proof methods ^We will discuss ten proof methods: 1.Direct proofs 2.Indirect proofs 3.Vacuous proofs 4.Trivial proofs 5.Proof by contradiction 6.Proof by cases 7.Proofs of equivalence 8.Existence proofs 9.Uniqueness proofs 10.Counterexamples

14. ch2.1 Sets CSci 2011

15. 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

16. CSci 2011 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”

17. CSci 2011 Specifying a set ^A 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}

18. CSci 2011 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

19. CSci 2011 The universal set 1  U is the universal set – the set of all of elements (or the “universe”) from which given any set is drawn  For the set {-2, 0.4, 2}, U would be the real numbers  For the set {0, 1, 2}, U could be the N, Z, Q, R depending on the context  For the set of the vowels of the alphabet, U would be all the letters of the alphabet

20. CSci 2011 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 the alphabet ^The individual elements are usually not written in a Venn diagram a ei ou b c df g hj kl m npq rst vwx yz U S

21. CSci 2011 Sets of sets ^Sets 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

22. CSci 2011 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

23. CSci 2011 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)

24. CSci 2011 ^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

25. CSci 2011 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

26. CSci 2011 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 2 n elements

27. CSci 2011 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)

28. CSci 2011 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 }

29. CSci 2011 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)

30. ch2.2 Set Operations CSci 2011

31. 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  = AIdentity law  A U U = U Domination law  A U A = AIdempotent law  A U B = B U ACommutative law  A U (B U C) = (A U B) U CAssociative law

32. CSci 2011 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 = AIdentity law  A ∩  =  Domination law  A ∩ A = AIdempotent law  A ∩ B = B ∩ ACommutative law  A ∩ (B ∩ C) = (A ∩ B) ∩ CAssociative law

33. CSci 2011 Disjoint sets ^Formal 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

34. CSci 2011 ^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

35. CSci 2011 Complement sets ^Formal definition for the complement of a set: A = { x | x  A } = A c  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  (A c ) c = AComplementation law  A U A c = U Complement law  A ∩ A c = Complement law

36. CSci 2011 Set identities A = A AU = A Identity Law AU = U A =  Domination law AA = A AA = A Idempotent Law (A c ) c = A Complement Law AB = BA AB = BA Commutative Law (AB) c = A c B c (AB) c = A c B c De Morgan’s Law A(BC) = (AB)C A(BC) = (AB)C Associative Law A(BC) = (AB)(AC) A(BC) = (AB)(AC) Distributive Law A(AB) = A A(AB) = A Absorption Law A  A c = U A  A c =  Complement Law

37. CSci 2011 How to prove a set identity ^For example: A ∩ B=B-(B-A) ^Four methods:  Use the basic set identities  Use membership tables  Prove each set is a subset of each other  Use set builder notation and logical equivalences

38. CSci 2011 What we are going to prove… A ∩ B=B-(B-A) AB A∩BA∩BB-AB-(B-A)

39. CSci 2011 Proof by Set Identities  A  B = A - (A - B) Proof) A - (A - B) = A - (A  B c ) = A  (A  B c ) c = A  (A c  B) = (A  A c )  (A  B) =   (A  B) = A  B

40. CSci 2011 Showing each is a subset of the others  (A  B) c = A c  B c Proof) Want to prove that (A  B) c  A c  B c and (A  B) c  A c  B c x  (A  B) c  x  (A  B)   (x  A  B)   (x  A  x  B)   (x  A)   (x  B)  x  A  x  B  x  A c  x  B c  x  A c  B c

41. CSci 2011 Examples ^Let A, B, and C be sets. Show that: a)(AUB)  (AUBUC) b)(A ∩ B ∩ C)  (A ∩ B) c)(A-B)-C  A-C d)(A-C) ∩ (C-B) = 