1. 1 Set Theory

2. 2 Set Properties Commutative Laws: Associative Laws: Distributive Laws:

3. 3 Set Properties Double Complement Law: De Morgan’s Laws: Absorption Laws:

4. 4 Showing that an alleged set property is false Statement: For all sets A,B and C, A  (B  C) = (A  B)  C. The following counterexample shows that the statement is false. Counterexample: Let A={1,2,3,4}, B={3,4,5,6}, C={3}. Then B  C = {4,5,6} and A  (B  C) = {1,2,3}. On the other hand, A  B = {1,2} and (A  B)  C = {1,2}. Thus, for this example A  (B  C) ≠ (A  B)  C.

5. 5 Empty Set The unique set with no elements is called empty set and denoted by . Set Properties that involve . For all sets A, 1.   A 2. A   = A 3. A   =  4. A  A c = 

6. 6 Disjoint Sets  A and B are called disjoint iff A  B = .  Sets A 1, A 2, …, A n are called mutually disjoint iff for all i,j = 1,2,…, n A i  A j =  whenever i ≠ j.  Examples: 1) A={1,2} and B={3,4} are disjoint. 2) The sets of even and odd integers are disjoint. 3) A={1,4}, B={2,5}, C={3} are mutually disjoint. 4) A  B, B  A and A  B are mutually disjoint.

7. 7 Partitions Definition: A collection of nonempty sets {A 1, A 2, …, A n } is a partition of a set A iff 1. A = A 1  A 2  …  A n 2. A 1, A 2, …, A n are mutually disjoint. Examples: 1) {Z +, Z -, {0} } is a partition of Z. 2) Let S 0 ={n  Z | n=3k for some integer k} S 1 ={n  Z | n=3k+1 for some integer k} S 2 ={n  Z | n=3k+2 for some integer k} Then {S 0, S 1, S 2 } is a partition of Z.

8. 8 Power Sets Definition: Given a set A, the power set of A, denoted P (A), is the set of all subsets of A. Example: P ({a,b}) = { , {a}, {b}, {a,b}}. Properties: 1) If A  B then P (A)  P (B). 2) If a set A has n elements then P (A) has 2 n elements.