1 Set Theory

Notation S={a, b, c} refers to the set whose elements are a, b and c. a  S means “a is an element of set S”. d  S means “d is not an element of set S”. {x  S | P(x)} is the set of all those x from S such that P(x) is true. E.g., T={x  Z | 0<x<10}. Notes: 1) {a,b,c}, {b,a,c}, {c,b,a,b,b,c} all represent the same set. 2) Sets can themselves be elements of other sets, e.g., S={ {Mary, John}, {Tim, Ann}, …}

3 Relations between sets Definition: Suppose A and B are sets. Then A is called a subset of B: A  B iff every element of A is also an element of B. Symbolically, A  B   x, if x  A then x  B. A  B   x such that x  A and x  B. B A A B B A A  BA  B

4 Relations between sets Definition: Suppose A and B are sets. Then A equals B: A = B iff every element of A is in B and every element of B is in A. Symbolically, A=B  A  B and B  A. Example: Let A = {m  Z | m=2k+3 for some integer k}; B = the set of all odd integers. Then A=B.

Operations on Sets Definition: Let A and B be subsets of a set U. 1. Union of A and B: A  B = {x  U | x  A or x  B} 2. Intersection of A and B: A  B = {x  U | x  A and x  B} 3. Difference of B minus A: B  A = {x  U | x  B and x  A} 4. Complement of A: A c = {x  U | x  A} Ex.: Let U=R, A={x  R | 3<x<5}, B ={x  R| 4<x<9}. Then 1) A  B = {x  R | 3<x<9}. 2) A  B = {x  R | 4<x<5}. 3) B  A = {x  R | 5 ≤x<9}, A  B = {x  R | 3<x ≤4}. 4) A c = {x  R | x ≤3 or x≥5}, B c = {x  R | x ≤4 or x≥9}

6 Properties of Sets  Theorem 1 (Some subset relations): 1) A  B  A 2) A  A  B 3) If A  B and B  C, then A  C.  To prove that A  B use the “element argument”: 1. suppose that x is a particular but arbitrarily chosen element of A, 2. show that x is an element of B.

7 Proving a Set Property Theorem 2 (Distributive Law): For any sets A,B and C: A  (B  C) = (A  B)  (A  C). Proof: We need to show that (I) A  (B  C)  (A  B)  (A  C) and (II) (A  B)  (A  C)  A  (B  C). Let’s show (I). Suppose x  A  (B  C) (1) We want to show that x  (A  B)  (A  C) (2)

8 Proving a Set Property Proof (cont.): x  A  (B  C)  x  A or x  B  C. (a) Let x  A. Then x  A  B and x  A  C  x  (A  B)  (A  C) (b) Let x  B  C. Then x  B and x  C. Thus, (2) is true, and we have shown (I). (II) is shown similarly (left as exercise).■

9 Set Properties Commutative Laws: Associative Laws: Distributive Laws:

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

11 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.

12 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 = 

13 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.

14 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.

15 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.