1、2.2: The Union Operator,For sets A, B, their union AB is the set containing all elements that are either in A, or (“”) in B (or, of course, in both). Formally, A,B: AB = x | xA xB. Note that AB is a superset of both A and B (in fact, it is the smallest such superset): A, B: (AB A) (AB B),a,b,c2,3 =

2、a,b,c,2,3 2,3,53,5,7 = 2,3,5,3,5,7 =2,3,5,7,Union Examples,The Intersection Operator,For sets A, B, their intersection AB is the set containing all elements that are simultaneously in A and (“”) in B. Formally, A,B: AB=x | xA xB. Note that AB is a subset of both A and B (in fact it is the largest su

3、ch subset): A, B: (AB A) (AB B),a,b,c2,3 = _ 2,4,63,4,5 = _,Intersection Examples,4,Disjointedness,Two sets A, B are calleddisjoint (i.e., unjoined)iff their intersection isempty. (AB=) Example: the set of evenintegers is disjoint withthe set of odd integers.,Set Difference,For sets A, B, the differ

4、ence of A and B, written AB, is the set of all elements that are in A but not B. Formally: A B : x xA xB x xA xB Also called: The complement of B with respect to A.,Set Difference Examples,1,2,3,4,5,6 2,3,5,7,9,11 = _ Z N , 1, 0, 1, 2, 0, 1, = x | x is an integer but not a nat. # = x | x is a negati

5、ve integer Result = , 3, 2, 1,1,4,6,Set Difference,AB is whats left after B“takes a bite out of A”,Set A,Set B,Symmetric Difference,For sets A, B, the symmetric difference of A and B, written A B, is the set:A B : (A B) (B A) (A B) (A B) E.g. A= 1, 2, 3, 4, 5, B = 4, 5, 6, 7, 8, A B= 1, 2, 3, 6, 7,

6、8,Set Complements,When the context clearly defines The universe of discourse U, we say that for any set AU, the complement of A, written , is the complement of A w.r.t. U, i.e., it is UA. E.g., If U=N, Note: Alternative notation is Ac, and x| xU and x A.,More on Set Complements,An equivalent definit

7、ion, when U is clear:,A,U,Set Identities,Identity: A = A = AU Domination: AU = U, A = Idempotent: AA = A = AA Double complement: Commutative: AB = BA, AB = BA Associative: A(BC)=(AB)C , A(BC)=(AB)C Distributive: ()=()() ()=()(),DeMorgans Law for Sets,Exactly analogous to (and provable from) DeMorgan

8、s Law for propositions. Dont memorize them, understand them!,Proving Set Identities,To prove statements about sets, of the form E1 = E2 (where the Es are set expressions), here are three useful techniques: 1. Prove E1 E2 and E2 E1 separately. 2. Use set builder notation & logical equivalences. 3. Us

9、e a membership table.,Method 1: Mutual subsets,Example: Show A(BC)=(AB)(AC). Part 1: Show A(BC)(AB)(AC). Assume xA(BC), & show x(AB)(AC). We know that xA, and either xB or xC. Case 1: xB. Then xAB, so x(AB)(AC). Case 2: xC. Then xAC , so x(AB)(AC). Therefore, x(AB)(AC). Therefore, A(BC)(AB)(AC). Par

10、t 2: Show (AB)(AC) A(BC). ,Method 2: Logical equivalence,Show or or is a tautology. But the consequent is always false. Therefore, the antecedent better always be false also.,Method 3: Membership Tables,Just like truth tables for propositional logic. Columns for different set expressions. Rows for a

11、ll combinations of memberships in constituent sets. Use “1” to indicate membership in the derived set, “0” for non-membership. Prove equivalence with identical columns.,Membership Table Example,Prove (AB)B = AB.,Membership Table Exercise,Prove (AB)C = (AC)(BC).,Generalized Unions & Intersections,Sin

12、ce union & intersection are commutative and associative, we can extend them from operating on ordered pairs of sets (A,B) to operating on sequences of sets (A1,An), or even on unordered sets of sets, X=A | P(A).,Generalized Union,Binary union operator: AB n-ary union:AA2An : (A1 A2) ) An)(grouping &

13、 order is irrelevant) “Big U” notation: Or for infinite sets of sets:,Generalized Intersection,Binary intersection operator: AB n-ary intersection:A1A2An(A1A2)An)(grouping & order is irrelevant) “Big Arch” notation: Or for infinite sets of sets:,Examples,Let,Examples,if Then Then,Inclusion/Exclusion

14、,Example: How many people are wearing a watch? How many people are wearing sneakers?,How many people are wearing a watch OR sneakers?,|A B| = |A| + |B| - |A B|,Generalized Inclusion/Exclusion,Suppose we have:,And I want to know |A U B U C|,|A U B U C| = |A| + |B| + |C|,+ |A B C|,- |A B| - |A C| - |B

15、 C|,Representations,A frequent theme of this course will be methods of representing one discrete structure using another discrete structure of a different type. E.g., one can represent natural numbers as Bit strings: 0:0, 1:1, 2:10, 3:11, 4:100, ,Representing Sets with Bit Strings,For an enumerable u.d. U with ordering x1, x2, , represent a finite set SU as the finite bit string B=b1b2bn wherei: xiS (in bi=1). e.g. U=N, S=2,3,5,7,11, B=001101010001. In this repr