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}, …}

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 A B B A  B A  B A  B

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: 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 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.

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)

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). ■

Set Properties Commutative Laws: Associative Laws: Distributive Laws:

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

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 .

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 = 

Disjoint Sets A and B are called disjoint iff A  B =  . Sets A1, A2, …, An are called mutually disjoint iff for all i,j = 1,2,…, n Ai  Aj =  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.

Partitions Definition: A collection of nonempty sets {A1, A2, …, An} is a partition of a set A iff 1. A = A1  A2  …  An 2. A1, A2, …, An are mutually disjoint. Examples: 1) {Z+, Z-, {0} } is a partition of Z. 2) Let S0={n  Z | n=3k for some integer k} S1={n  Z | n=3k+1 for some integer k} S2={n  Z | n=3k+2 for some integer k} Then {S0, S1, S2} is a partition of Z.

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 2n elements.