Discrete Mathematics I Lectures Chapter 6

1. Discrete Mathematics ILectures Chapter 6 Some material adapted from lecture notes provided by Dr. Chungsim Han and Dr. Sam Lomonaco Dr. Adam Anthony Spring 2011

2. … and now for something completely different… Set Theory Actually, you will see that logic and set theory are very closely related.

3. Set Theory • Set: Collection of objects (“elements”) • aA“a is an element of A” “a is a member of A” • aA“a is not an element of A” • A = {a1, a2, …, an} “A contains…” • Order of elements is meaningless • It does not matter how often the same element is listed – repeats are OK!.

4. Set Equality • Sets A and B are equal if and only if they contain exactly the same elements. • Examples: • A = {9, 2, 7, -3}, B = {7, 9, -3, 2} : A = B • A = {dog, cat, horse}, B = {cat, horse, squirrel, dog} : A  B • A = {dog, cat, horse}, B = {cat, horse, dog, dog} : A = B

5. Examples for Sets • “Standard” Sets: • Natural numbers N = {0, 1, 2, 3, …} • Integers Z = {…, -2, -1, 0, 1, 2, …} • Positive Integers Z+ = {1, 2, 3, 4, …} • Real Numbers R = {47.3, -12, , …} • Rational Numbers Q = {1.5, 2.6, -3.8, 15, …}(correct definition will follow)

6. Examples for Sets • A = “empty set/null set” • A = {z} Note: zA, but z  {z} • A = {{b, c}, {c, x, d}} • A = {{x, y}} Note: {x, y} A, but {x, y}  {{x, y}} • A = {x | P(x)}“set of all x such that P(x)” • A = {x | xN x > 7} = {8, 9, 10, …}“set builder notation”

7. Examples for Sets • We are now able to define the set of rational numbers Q: • Q = {a/b | aZ bZ+} or • Q = {a/b | aZ bZ b0} • And how about the set of real numbers R? • R = {r | r is a real number}That is the best we can do.

8. Exercise 1 • What are the members of the following sets? • A = {x| x  R  x2 = 5} • B = {n | n  Z  n  2} • Use set-builder notation to describe the following sets: • The set of all positive integers that are divisible by 3 • The set of all animals with black and white stripes

9. Why Sets? • Sets add another layer of notational convenience • Call it Logic 3.0! • We can use sets to simplify logic expressions • Let BW = {x | x is an animal and x has black and white stripes} • All animals with black and white stripes are easy to see •  a  BW, easy-to-see(a) • There exists an animal without black and white stripes that is easy to see • aBW, easy-to-see(a)

10. Subsets • A  B“A is a subset of B” • A  B if and only if every element of A is also an element of B. • We can completely formalize this: • A  B  x (xA  xB) • Examples: A = {3, 9}, B = {5, 9, 1, 3}, A  B ? true A = {3, 3, 3, 9}, B = {5, 9, 1, 3}, A  B ? true A = {1, 2, 3}, B = {2, 3, 4}, A  B ? false

11. U B C A Subsets • Useful rules: • A = B  (A  B)  (B  A) • (A  B)  (B  C)  A  C (see Venn Diagram)

12. Subsets • Useful rules: •   A for any set A • A  A for any set A • Proper subsets: • A  B “A is a proper subset of B” • A  B  x (xA  xB)  x (xB  xA) • or • A  B  x (xA  xB)  x (xB  xA)

13. Exercise 2 Let A = {1,3,4}, B = {1,2,4}, C = {4,3,1,1}, D = {1,2,3,4} • Is 3A? • Is {3}  A? • Is {3}  A? • Is 3  A? • Is 3  B? • Is 3  B? • Is A  B? • Is A  D? • Is C  A? • Is D  A? • Is A = C? • Is A = D?

14. Exercise 3 • Let A = {a | a  Z    a  -3}, B = {b | b  Z  bk = 5 for some k  Z+}, C = {c | c  Z  c is prime  7 < c < 10} • Describe A, B and C by listing their elements • Is B  A? • Is A  B? • Is C  A?

15. Cardinality of Sets • If a set S contains n distinct elements, nN,we call S a finite set with cardinality n. • Examples: • A = {Mercedes, BMW, Porsche}, |A| = 3 B = {1, {2, 3}, {4, 5}, 6} |B| = 4 C =  |C| = 0 D = { xN | x  7000 } |D| = 7001 E = { xN | x  7000 } E is infinite!

16. The Power Set • (A) “power set of A” • (A) = {B | B  A} (contains all subsets of A) • Examples: • A = {x, y, z} • (A)= {, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}} • A =  • (A) = {} • Note: |A| = 0, | (A)| = 1

17. A 1 2 3 4 5 6 7 8 x x x x x x x x x y y y y y y y y y z z z z z z z z z The Power Set • Cardinality of power sets: • | (A) | = 2|A| • Imagine each element in A has an “on/off” switch • Each possible switch configuration in A corresponds to one element in 2A • For 3 elements in A, there are 222 = 8 elements in P(A)

18. Exercise 4 • Find the power sets ({1,2}) and ({1,2,3})

19. Ordered Pairs • We are familiar with ordered pairs in algebra when we plot on a Cartesian coordinate system: • The pair (x,y) is orderedso that we know how to plot it on the axes • Important: (a,b) = (c,d)  a = c and b = d • The order matters! (x,y)

20. Cartesian Product • The ordered n-tuple(a1, a2, a3, …, an) is an ordered collection of objects. • Just a general version of the ordered pair: • Two ordered n-tuples (a1, a2, a3, …, an) and (b1, b2, b3, …, bn) are equal if and only if they contain exactly the same elements in the same order, i.e. ai = bi for 1  i  n. • The Cartesian product of two sets is defined as: • AB = {(a, b) | aA  bB} • Example: A = {x, y}, B = {a, b, c}AB = {(x, a), (x, b), (x, c), (y, a), (y, b), (y, c)}

21. Cartesian Product • The Cartesian product of two sets is defined as: AB = {(a, b) | aA  bB} • Example: • A = {good, bad}, B = {student, prof} • AB = { (bad, prof)} (good, student), (good, prof), (bad, student), (prof, bad)} BA = { (student, good), (prof, good), (student, bad),

22. Cartesian Product • Note that: • A =  • A =  • For non-empty sets A and B: AB  AB  BA • |AB| = |A||B| • The Cartesian product of two or more sets is defined as: • A1A2…An = {(a1, a2, …, an) | aiA for 1  i  n}

23. Set Operations • Union: AB = {x | xA  xB} • Example: A = {a, b}, B = {b, c, d} AB = {a, b, c, d} • Intersection: AB = {x | xA  xB} • Example: A = {a, b}, B = {b, c, d} AB = {b}

24. Set Operations • Two sets are called disjoint if their intersection is empty, that is, they share no elements: • AB =  • If a set C = A  B and A and B are disjoint, then the set P = {A,B} is referred to as a partition of C • The difference between two sets A and B contains exactly those elements of A that are not in B: • A-B = {x | xA  xB}Example:A = {a, b}, B = {b, c, d}, A-B = {a}

25. Set Operations • Given a Universe identified by the set U, the complement of a set A contains exactly those elements under consideration that are not in A: • Ac = U-A • Example:U = N, B = {250, 251, 252, …} Bc = {0, 1, 2, …, 248, 249}

26. Exercise 5 • Let the Universe U = {0,1,2,3,…,9} and: A = {0,2,4,6,8}B = {1, 2, 4, 7}C = {1,5,7} • Find the following: • A  B • A  B • Bc • A - B • (A B)c • A  (B C) • (A  B) C • Ac Bc

27. Exercise 6 • Let A = {1,2,3,4,5,6,7} Which collections form a partition of A? • P1 = {1,3}, {2,5}, {4,6,7} • P2 ={1}, {2,4,6}, {3,5} • P3 ={1,3,5,7}, {2,4,6}, {3,6} • Let B = {x | x  R  x  -2 }, C = {x | x  R  -2 <x  4}, D = {x | x  R  x > 4} • Is P4 = {B,C,D} a partition of R?

28. Venn Diagrams of Operations • NOTE: The amount that A and B overlap is not always guaranteed A B

29. Venn Diagrams of Operations • A A B

30. Venn Diagrams of Operations • B A B

31. Venn Diagrams of Operations • A  B A B

32. Venn Diagrams of Operations • A  B

33. Venn Diagrams of Operations • A - B

34. Venn Diagrams of Operations • Ac

35. Exercise 7 • Draw a Venn Diagram to represent the following expressions over the sets A,B and C: • A  B =  • A  B = A and C  B • C  A  B • Ac – C • A  B   and A  C   and B  C   BUT A  B  C  

36. Section 6.2,6.3: Properties of Sets • Unifying Logic and Sets • Proofs with sets • Set identities

37. Logic and Sets • We can interpret our set operations logically: • A  B  x  A  x  B (Set-based implication) • A  B  x  A  x  B (Set-based OR) • A  B  x  A  x  B (Set-based AND) • x  Ac x  A (Set-based Negation) • A – B  x  A  x  B (NEW expressive power!) • (x,y)  A x B  x  A  y  B (NEW expressive power!)

38. Element Arguments • Prove that for all sets A and B, A  B  A • Normal starting Point: “Suppose A and B are arbitrarily chosen sets” • Where does this get us? • We make more progress using an ‘element argument’: • Note that A  B  A translates to “If x  A  B then x  A” • From this, we get the statement: “Suppose A and B are arbitrarily chosen sets and x  A  B” • Idea: focusing on a single arbitrary element in a set lets us prove something about every single element in the set!

39. Transitivity Revisited • Prove, using logical interpretations, the transitive property for subsets: • For all sets A, B and C, if A  B and B  C, then A  C.

40. Proving Equality • How can we Prove A(BC) = (AB)(AC)? • Recall: A = B  A  B  B  A • To prove A = B, you must show both subset claims are true • So an equality proof really requires 2 element proofs!

41. Prove A(BC) = (AB)(AC). • Prove that A(BC)  (AB)(AC): Suppose that x is an arbitrarily chosen element of A(BC). There are 2 cases for x: x  A or x  (BC)Case 1: x  ASince x  A, by definition of union, x  (A  B) and x  (A  C). Therefore, by definition, x  (AB)(AC). Case 2: x  (BC). By definition of intersection, x  B and x  C. By definition of union, x  (AB) and x  (AC). Therefore, by definition of intersection, x  (AB)(AC)

42. Prove A(BC) = (AB)(AC). • Prove that (AB)(AC)  A(BC) • Drawing a Venn diagram might help us think this through… Suppose that x is an arbitrarily chosen element of (AB)(AC). We’ll analyze this one element in two cases: x  A and x  A. Case 1: x  A It immediately follows that x  A(BC) by definition of union. Case 2: x  A Since x  (AB)(AC), it must be true that x  (AB) and x  (AC). Since x  A in this case, it must follow that x  B and x  C or else x would not be a member of (AB)(AC). Therefore, by definition of intersection, x  (BC) and by definition of Union, x  A(BC)

43. OR…Use logic instead! • How can we prove A(BC) = (AB)(AC)? • Suppose x is an arbitrary element of A(BC). Then by definition, the following statements are true: x(A(BC)) • (xA)  (x(BC)) • (xA)  (xB  xC) • (xA  xB)  (xA  xC)(distributive law for logical expressions) • (x(AB))  (x(AC)) • x((AB)(AC)) Since each statement is a biconditional with the one before it, x(A(BC))  x((AB)(AC)) and x((AB)(AC))  x(A(BC)) . So by definition, (A(BC))  ((AB)(AC)) and ((AB)(AC))  (A(BC)). Therefore, A(BC) = (AB)(AC)

44. Which method to use? • You may use an element argument or logic conversions to prove set identities on assignments and exams • Which one you choose depends on your strengths • Neither approach is ‘easier’ than the other • Element arguments provide more direction about where you’re headed • Logic arguments have less direction, but tend to be more concise

45. Set Identities—Look Familiar?

46. Exercise 1 • Use Venn Diagrams to verify the following set-theoretic DeMorgan’s law: • (A  B)c = Ac Bc

47. Exercise 2 • Recall that two sets are disjoint if their intersection is the empty set (A  B = ). • Draw venn diagrams for the sets A – B and B – A • Use Theorem 6.2.2 to construct a proof that A – B and B – A are disjoint

48. Exercise 3 • Use Theorem 6.2.2 to prove the following: (A  (B  C)  ((A  B) – C) = A  B

49. A Technology Tie-In • Don’t forget your set theory—You’ll use it in Databases! • Each ‘table’ in a database is called a relation, filled with tuples • Database theory says that a relation is actually a subset of the universe of all possible tuples Sailors

50. A Technology Tie-In • SQL (structured query language): Select SID, SNAME, RATING, AGE FROM SAILORS WHERE AGE > 40 • S = {X  SAILORS | AGE > 40} • See the similarity? Remembering set-theory will really help you work with databases Sailors