1 / 15

Set Theory

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 } .

ollie
Télécharger la présentation

Set Theory

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Set Theory

  2. 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. B A 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  B A  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.

  5. 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}

  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} andA(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  Ac = 

  13. 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.

  14. 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.

  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 thenP (A) P (B) . 2)If a set A has n elements then P (A) has 2n elements.

More Related