Set Theory 1 Notation Sa b c refers

Set Theory 1

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}, …}

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. A A B B A B A A B B 3

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=2 k+3 for some integer k}; B = the set of all odd integers. Then A=B. 4

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: Ac = {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) Ac = {x R | x ≤ 3 or x≥ 5}, Bc = {x R | x ≤ 4 or x≥ 9}

Properties of Sets Ø Theorem 1 (Some subset relations): 1) A B A 2) A A B 3) If A B and B C, then A C. Ø 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. 6

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) 7

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). ■ 8

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

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

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. 11

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 Ac = 12

Disjoint Sets Ø A and B are called disjoint iff A B = . Ø Sets A 1, A 2, …, An are called mutually disjoint iff for all i, j = 1, 2, …, n Ai Aj = whenever i ≠ j. Ø 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. 13

Partitions • Definition: A collection of nonempty sets {A 1, A 2, …, An} is a partition of a set A iff 1. A = A 1 A 2 … An 2. A 1, A 2, …, An are mutually disjoint. • Examples: 1) {Z+, Z-, {0} } is a partition of Z. 2) Let S 0={n Z | n=3 k for some integer k} S 1={n Z | n=3 k+1 for some integer k} S 2={n Z | n=3 k+2 for some integer k} Then {S 0, S 1, S 2} is a partition of Z. 14

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. 15