1. Set

2. Outline Universal Set Venn Diagram Operations on Sets

3. Outline Partitions of Set Finite and Infinite Sets Set Cardinality Cartesian Product Set Identities

4. Universal Set The set of all elements under consideration is called the Universal Set. The Universal Set is usually denoted by U and is represented by a rectangle. U

5. Venn Diagram Venn diagrams are often used to indicate the relationships between sets. The Universal Set is represented by the interior of a rectangle, and the other sets are represented by circle within the rectangle.

6. Venn Diagram Draw a Venn Diagram that represents V, the set of vowels in the English Alphabet We draw a rectangle to indicate the universal set U, which is the set of the 26 letters of the English alphabet. Inside this rectangle we draw a circle to represent V. Inside this circle we indicate the elements of V with points V

7. Venn Diagram Example –A  B –Venn Diagram Showing that A is a Subset of B. A B

8. Venn Diagram Example –A  B –Venn Diagram Showing that A is a Subset of B. A=B

9. Venn Diagram –A  B –Venn Diagram Showing that A is not a Subset of B. B A AB

10. Venn Diagram –A  B –Venn Diagram Showing that A is not a Subset of B. AB

11. Operations on Sets Union –Let A and B be subsets of a universal set U. –The union of A and B, denoted A U B, is the set of all elements that are in at least one of A or B. –A U B = {x  U | x  A or x  B} –For any set S, S U  = S and S U U = U

12. Operations on Sets Example Let –U = {a, b, c, d, e, f, g} –A = {a, c, e, g} –B = {d, e, f, g} Then –A U B = {a, c, d, e, f, g} AB

13. Operations on Sets Intersection –Let A and B be subsets of a universal set U. –The intersection of A and B, denoted A ∩ B, is the set of all elements that are common to both A and B. –A ∩ B = {x  U | x  A and x  B} –For any set S, S ∩  =  and S ∩ U = S

14. Operations on Sets Example Let –U = {a, b, c, d, e, f, g} –A = {a, c, e, g} –B = {d, e, f, g} Then –A ∩ B = {e, g} AB

15. Operations on Sets Difference –Let A and B be subsets of a universal set U. –The difference of A minus B (or relative complement of A in B), denoted A − B, is the set of all elements that are in A and not B. –A – B = {x  U | x  A and x  B} –For any set S, S –  = S and S – U = 

16. Operations on Sets Example Let –U = {a, b, c, d, e, f, g} –A = {a, c, e, g} –B = {d, e, f, g} Then –A – B = {a, c} AB Take note that in set Difference, A – B is different from B – A.

17. Operations on Sets Example Let –U = {a, b, c, d, e, f, g} –A = {a, c, e, g} –B = {d, e, f, g} Then –B – A = {d, f} AB

18. Operations on Sets Symmetric Difference –Let A and B be subsets of a universal set U. –The symmetric difference of A and B, denoted A  B, is the set of all elements that are in either A or B but not both or the union of the difference of A and B and B and A. –A  B = {x  U | x  A and x  B} U {x  U | x  B and x  A} –A  B = (A-B) U (B-A)

19. Operations on Sets Example Let –U = {a, b, c, d, e, f, g} –A = {a, c, e, g} –B = {d, e, f, g} Then –A  B = {a, c, d, f} AB B

20. Operations on Sets Complement –Let A be a subset of a universal set U. –The complement of A, denoted A c, is the set of all elements in U that are not in A. –A c = {x  U | x  A} –Empty sets and universal set are related by  C =U and U c = 

21. Operations on Sets Example Let –U = {a, b, c, d, e, f, g} –A = {a, c, e, g} Then –A c = {b, d, f} A AcAc

22. Exercise Let A = {1, 3, 5, 7, 9}, B = {3, 6, 9}, and C = {2, 4, 6, 8}. Find each of the following and draw a Venn Diagram representing each operation. 1.B – A6. B  C 2.A U B7. A  B 3.A – B8. A U C 4.A ∩ B9. A U C U C 5.B U C10. A – C

23. Solution to Exercise Let –A = {1, 3, 5, 7, 9}, –B = {3, 6, 9} Then –B – A = {6} AB

24. Solution to Exercise Let –A = {1, 3, 5, 7, 9}, –B = {3, 6, 9} Then –A U B = {1, 3, 5, 7, 9} AB

25. Solution to Exercise Let –A = {1, 3, 5, 7, 9}, –B = {3, 6, 9} Then –A – B = {1, 5, 7} AB

26. Solution to Exercise Let –A = {1, 3, 5, 7, 9}, –B = {3, 6, 9} Then –A ∩ B = {3, 9} AB

27. Solution to Exercise Let –B = {3, 6, 9} –C = {2, 4, 6, 8} Then –B U C = {2, 3, 4, 6, 8, 9} BC

28. Solution to Exercise Let –B = {3, 6, 9} –C = {2, 4, 6, 8} Then –B  C = {2, 3, 4, 8} BB C

29. Solution to Exercise Let –A = {1, 3, 5, 7, 9} –B = {3, 6, 9} Then –A  B = {1, 5, 6, 7} AB B

30. Solution to Exercise Let –A = {1, 3, 5, 7, 9} –C = {2, 4, 6, 8} Then –A U C = {1, 2, 3, 4, 5, 6, 7, 8, 9} AC

31. Solution to Exercise Let –A = {1, 3, 5, 7, 9} –C = {2, 4, 6, 8} Then –A U C U C = {1, 2, 3, 4, 5, 6, 7, 8, 9} C A C

32. Solution to Exercise Let –A = {1, 3, 5, 7, 9}, –C = {2, 4, 6, 8} Then –A – C = {1, 3, 5, 7, 9} AC

33. Partitions of Set In many applications of set theory, sets are divided up into non-overlapping (or disjoint) pieces. Such a division is called a partition. Two sets are called disjoint if, and only if, they have no elements in common. A ∩ B = 

34. Partitions of Set Let –A = {1, 3, 5} –B = {2, 4, 6} Then –A ∩ B = {1, 3, 5} ∩ {2, 4, 6} –A ∩ B =  –A and B are disjoint since they have no elements in common.

35. Power Set There are various situations in which it is useful to consider the set of all subsets of a particular set. The power set axiom guarantees that this is a set. Given a set A, the power set of A, denoted by P(A), is the set of all subsets of A.

36. Power Set Example Let –A = {1, 2} then –P(A) = { , {1}, {2}, {1, 2}} If A has n elements then P(A) has 2 n elements.

37. Exercise Determine the power set of the following sets. 1.A={a, b, c, d} 2.B={1, 2, 3, 4,5} 3.C={5, 10, 15}

38. Solution to Exercise Determine the power set of the following sets. A={a, b, c, d} P(A) = { ∅, {a}, {b}, {c}, {d}, {a,b}, {a,c}, {a,d}, {b,c}, {b,d}, {c,d}, {a,b,c}, {a,b,d}, {a,c,d}, {b,c,d}, {a,b,c,d} }

39. Solution to Exercise Determine the power set of the following sets. B={1, 2, 3, 4, 5} P(B) = { ∅, { 1 }, { 2 }, { 3 }, { 4 }, { 5 }, { 1, 2 }, { 1, 3 }, { 1, 4 }, { 1, 5 }, { 2, 3 }, { 2, 4 }, { 2, 5 }, { 3, 4 }, { 3, 5 }, { 4, 5 }, { 1, 2, 3 }, { 1, 2, 4 }, { 1, 2, 5 }, { 1, 3, 4 }, { 1, 3, 5 }, { 1, 4, 5 }, { 2, 3, 4 }, { 2, 3, 5 }, { 2, 4, 5 }, { 3, 4, 5 }, { 1, 2, 3, 4 }, { 1, 2, 3, 5 }, { 1, 2, 4, 5 }, { 1, 3, 4, 5 }, { 2, 3, 4, 5 }, { 1, 2, 3, 4, 5 } }

40. Solution to Exercise Determine the power set of the following sets. C={5, 10, 15} P(C) = { ∅, {5}, {10}, {15}, {5,10}, {5,15}, {10,15}, {5,10,15}}

41. Finite and Infinite Sets A set is called finite if, and only if, it is the empty set or there is one-to-one correspondence from {1,2,3, …, n} to it, where n is a positive integer. A non empty set that cannot be put into one-to- one correspondence with {1,2,3,…,n}, for any positive integer n, is called infinite set.

42. Set Cardinality The number of elements in a finite set A is termed as the cardinality of set A, denoted by |A|. Thus, if A = {1, 2, 3} then |A| = 3. Let A and B be any sets. A has the same cardinality as B if, and only if, there is a one-to- one correspondence from A to B(Cardinality means “the total number of elements in a set”). When A and B have the same cardinality, we write |A| = |B|.

43. Exercise Determine the cardinality of the following sets. 1.A={1} 2.B={1, 3, 5, 7} 3.C={a,b,c,d,e,f,g} 4.D={a,b,c,d,…,z}

44. Solution to Exercise Determine the cardinality of the following sets. 1.A={1}|A|=1 2.B={1, 3, 5, 7}|B|=4 3.C={a,b,c,d,e,f,g}|C|=6 4.D={a,b,c,d,…,z}|D|=26

45. Cartesian Product The order of elements in a collection is often important. Because sets are unordered, a different structure is needed to represent ordered collections. This is provided by ordered n-tuples. The ordered n-tuple (a 1, a 2,..., a n ) is the ordered collection that has a 1 as its first element, a 2 as its second element,..., and a n as its nth element.

46. Cartesian Product Cartesian Product of Sets Given sets A 1, A 2,..., A n, the Cartesian product of A 1, A 2,..., A n denoted A 1 × A 2 ×... × A n, is the set of all ordered n-tuples (a 1, a 2,..., a n ) where a 1 ∈ A 1, a 2 ∈ A 2,..., a n ∈ A n. –A 1 × A 2 ×· · ·× A n = {(a 1, a 2,..., a n ) | a 1 ∈ A 1, a 2 ∈ A 2,..., a n ∈ A n }.

47. Cartesian Product Let A and B be sets. The Cartesian product of A and B, denoted A x B (read “A cross B”) is the set of all ordered pairs (a, b), where a is in A and b is in B. A x B = {(a, b)| a  A and b  B} If set A has m elements and set B has n elements then A x B has m x n elements.

48. Cartesian Product Example Let –A = {1, 2} –B = {a, b, c} Then –A x B = {(1,a), (1,b), (1,c), (2,a), (2, b), (2, c)} –B x A = {(a,1), (a,2), (b, 1), (b, 2), (c, 1), (c, 2)}

49. Cartesian Product A x B  B x A for non-empty and unequal sets A and B. A x  =  x A =  | A x B| = |A| x |B|

50. Exercise Let –A = {x, y} –B = {1, 2, 3} –C = {a, b} Find the Cartesian Product of the following: 1.A x B 2.A x B x C 3.A x C 4.B x A

51. Solution to Exercise Let –A = {x, y} –B = {1, 2, 3} –C = {a, b} 1.A x B={x1, x2, x3, y1, y2, y3} 2.A x B x C={x1a, x2a, x3a, x1b, x2b, x3b, y1a, y1b, y2a, y2b, y3a, y3b} 3.A x C={xa, xb, ya, yb} 4.B x A={1x, 1y, 2x, 2y, 3x, 3y}

52. Set Identities An identity is an equation that is universally true for all elements in some set. Example, –The equation a + b = b + a is an identity for real numbers because it is true for all real numbers a and b.

53. Set Difference Let A, B, and C be sets Idempotent Laws –A U A = A –A ∩ A = A Commutative Laws –A U B = B U A –A ∩ B = B ∩ A Associative Laws –(A U B) U C = A U (B U C) –(A ∩ B) ∩ C = A ∩ (B ∩ C) Distributive Laws –A ∩ (B U C) = (A ∩ B) U (A ∩ C) –A U (B ∩ C) = (A U B) ∩ (A U C) De Morgan’s –(A U B)’ = A‘ ∩ B’ –(A ∩ B)’ = A‘ U B’ Double Complement Law –(A’)’ = A Set Difference –A – B = A ∩ B’

54. Set Identities Example Prove that for all sets A, B, and C, –A − (B ∩ C) = (A − B) U (A − C) Solution 1.A − (B ∩ C) = A ∩ (B ∩ C)’Set Difference 2.A ∩ (B ∩ C) = A ∩ (B’ U C’)De Morgan’s 3.A ∩ (B’ U C’) = (A ∩ B’) U (A ∩ C’) Distributive 4.(A ∩ B’) U (A ∩ C’) = (A – B) U (A – C) set difference

55. Exercise Prove that for all sets A, B, and C, –(A U ( B ∩ C))’ = (C’ U B’) ∩ A’

56. Solution to Exercise Prove that for all sets A, B, and C, –(A U (B ∩ C))’ = (C’ U B’) ∩ A’ –(A U (B ∩ C))’ = A’ ∩ (B ∩ C)’ De Morgan’s Law –A’ ∩ (B’ ∩ C)’ = A’ ∩ (B’ U C’)De Morgan’s Law –A’ ∩ (B’ U C’) = (B’ U C’) ∩ A’ Commutative Law –(B’ U C’) ∩ A’ = (C’ U B’) ∩ A’Commutative Law

57. Summary The set of all elements under consideration is called the Universal Set. The Universal Set is usually denoted by U and is represented by a rectangle. Venn diagrams are often used to indicate the relationships between sets. The Universal Set is represented by the interior of a rectangle, and the other sets are represented by circle within the rectangle.

58. Summary The Operations on Sets are Union, Intersection, Difference, Set Difference, and Complement. Sets are divided up into non-overlapping (or disjoint) pieces. Two sets are called disjoint if, and only if, they have no elements in common. Power set is the set of all subsets of a particular set.

59. Summary A set is called finite if, and only if, it is the empty set or there is one-to-one correspondence from {1,2,3, …, n} to it, where n is a positive integer. A non empty set that cannot be put into one-to- one correspondence with {1,2,3,…,n}, for any positive integer n, is called infinite set. The number of elements in a finite set A is termed as the cardinality of set A, denoted by |A|.

60. Summary The Cartesian product of A and B, denoted A x B (read “A cross B”) is the set of all ordered pairs (a, b), where a is in A and b is in B. An identity is an equation that is universally true for all elements in some set. The basic set identities are Idempotent Laws, Commutative Laws. Associative Laws, Distributive Laws, De Morgan’s, Double Complement Law, and Set Difference.

61. QUESTION?