1. Fr: Discrete Mathematics by Washburn, Marlowe, and Ryan

2. - a set is a collection of objects - it can be denoted by letters (i.e. A, B, C,…) or by letters with subscripts (i.e. A 1, A 2, A 3,…)

3. The objects that a set contains are its elements. The notation a Є A means that a is an element of set A, and the notation a Є A means that a is not an element of the set A.

4. A set can be described by putting its elements in curly brackets (braces). Ex: the set whose elements are the integers 1, 2, 3 is denoted by { 1, 2, 3 }

5. A set can be described by giving a logical condition which its elements must satisfy. The set of elements x which satisfy P(x) is denoted A = { x : P(x) } This is called the set-builder notation

6. For example, the set A = { 1, 2, 3 } can be obtained using set-builder notation by taking P(x) to be the condition “x is an integer greater than or equal to 1 and less than or equal to 3”.

7.  A set is determined by its elements. Two sets A and B are equal if and only if they have the same elements. Ex: A = { 1, 2, 3, 4 } B = { 1, 2, 3, 4 }

8. Two sets A and B are equivalent if they have the same cardinality, that is ∣ A ∣ = ∣ B ∣ (cardinality refers to the number of elements) Ex:A = { 1, 2, 3 } B = { 3, 6, 7 }

9. Since sets are determined by their elements, there is a unique set ɸ which has no elements. ɸ is called the empty set or the null set.

10. A set is finite if its elements can be listed by a list that stops. A set which is not finite is infinite. The set A = { 1, 2, 3, …. n } is a finite set with n elements. The set of natural numbers N = { 1, 2, 3, …. } is an infinite set

11. Let A and B be sets. B is a subset of A if each element of B is an element of A. The notation B ⊂ A means that B is a subset of A.

12. If A is a set, the set of all subsets of A denoted by P(A) is called the powerset of A. Ex: A = { 1, 2 }, describe P(A) P(A) = { ɸ, {1}, {2}, {1, 2} }

13. If A and B are sets, then the product set A x B, is the set of ordered pairs (a, b), where a Є A and b Є B. In set-builder notation, A x B = { (a, b) ∣ a Є A and b Є B } Ex: A = { 1, 3 }B = { 2, 5 } A x B = { (1, 2), (1, 5), (3, 2), (3,5) }

14. The union of two subsets A ⊂ X and B ⊂ X : A ∪ B = { x Є X ∣ x Є A or x Є B } Ex:A = { 1, 3, 5 } B = { 2, 4, 6 } A ∪ B = { 1, 2, 3, 4, 5, 6 }

15. The intersection of the two subsets A ⊂ X and B ⊂ X: A ∩ B = { x Є X ∣ x Є A and x Є B } Ex:A = { 1, 3, 5 } B = { 2, 4, 6 } C = { 1, 2, 3 } A ∩ C = { 1, 3 }B ∩ C = {2 } A ∩ B = { } or ɸ Note: A and B are disjoint sets

16. The difference of two subsets A ⊂ X and B ⊂ X: A - B = { x Є X ∣ x Є A and x Є B } Ex: A = { 1, 3, 5, 6 } B = { 1, 2, 3, 4 } A - B = { 5, 6 }

17. The complement of a subset A ⊂ X: A c (or A’) = { x Є X ∣ x Є A } Ex : X = { 1, 2, 3, 4, 5 } A = { 1, 3, 5 } A’ = { 2, 4 }

18. A ∪ B (all of the yellow color) A – B (blue portion) AB AB

19. A ∩ B (shaded portion) A AB A’ (shaded portion)

20. I. Given: X = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } A = { 1, 5, 9 } B = { 2, 3, 6, 10 } Find: 1. A x B2. B x A 3. A ∪ B 4. A ∩ B 5. A – B6. B – A 7. A’8. B’ 9. P(A)10. P(B)

21. II. If A, B, and C are subsets of X, describe the following subsets with Venn diagrams. 1. ( A ∪ B ) - C 2. ( A ∩ B ) - C 3. ( A - B ) - C 4. A ∩ ( B ∪ C ) 5. ( A ∩ B ) ∪ ( A ∩ C )