Discrete Mathematics Set Operations

Transcription

1 Discrete Mathematics 1-3. Set Operations

2 Introduction to Set Theory A setis a new type of structure, representing an unordered collection (group, plurality) of zero or more distinct (different) objects. Set theory deals with operations between, relations among, and statements about sets. Sets are ubiquitous in computer software systems. Allof mathematics can be defined in terms of some form of set theory (using predicate logic). Discrete Mathematics, Spring

3 Basic notations for sets For sets, we ll use variables S, T, U, We can denote a set Sin writing by listing all of its elements in curly braces: {a, b, c} is the set of whatever 3 objects are denoted by a, b, c. Setbuilder notation: For any proposition P(x) over any universe of discourse, {x P(x)} is the set of all x such that P(x). Discrete Mathematics, Spring

4 Basic properties of sets Sets are inherently unordered: No matter what objects a, b, and c denote, {a, b, c} = {a, c, b} = {b, a, c} = {b, c, a} = {c, a, b} = {c, b, a}. All elements are distinct(unequal); multiple listings make no difference! If a=b, then {a, b, c} = {a, c} = {b, c} = {a, a, b, a, b, c, c, c, c}. This set contains at most 2 elements! Discrete Mathematics, Spring

5 Infinite Sets Conceptually, sets may be infinite(i.e., not finite, without end, unending). Symbols for some special infinite sets: N= {0, 1, 2, } The Natural numbers. Z= {,-2,-1, 0, 1, 2, } The Zntegers. R= The Real numbers, such as Infinite sets come in different sizes! Discrete Mathematics, Spring

6 Empty Set Definition: A set which does not contain any elements is an empty set, denoted by or {} or {x false} Example: x: x Discrete Mathematics, Spring

7 Subset and Superset Definition: Let S and T be any two sets. S is a subsetof T and T is a supersetof S, denoted by S T, if and only if every element of Sis an element of T, i.e., ( x)((x S) (x T)). Example S, S S. Discrete Mathematics, Spring

8 Set Equality Definition: Let Sand Tbe any two sets. Sand Tare declared to be equal if and only ifthey contain exactly the sameelements,i.e. S=T iff(s T) (S T) Note that it does not matter how the set is defined or denoted. Example: The set {1, 2, 3, 4} = {x xis an integer where x>0 and x<5 } = {x xis a positive integer whose square is >0 and <25} Discrete Mathematics, Spring

9 Proper Subsets & Supersets Definition: Let Sand Tbe any two sets. Sis a proper subset of T(Tis a proper superset of S), denoted by S T iffs T and S T. S T Venn Diagram equivalent of S T Example: {1,2} {1,2,3} Discrete Mathematics, Spring

10 Sets Are Objects, Too! The objects that are elements of a set may themselves be sets. Example: Let S={x x {1,2,3}} then S={, {1},{2},{3}, {1,2},{1,3},{2,3}, {1,2,3}} Note that 1 {1} {{1}}!!!! Discrete Mathematics, Spring

11 Basic Set Relations: Member of Definition: x S( xis in S )is the proposition that object xis an lementor memberof set S. e.g.3 N, a {x xis a letter of the alphabet} Can define set equality in terms of relation: S,T: S=T ( x: x S x T) Two sets are equal iffthey have all the same members. x S: (x S) xis not in S Discrete Mathematics, Spring

12 Cardinality and Finiteness S (read the cardinalityof S ) is a measure of how many different elements S has. E.g., =0, {1,2,3} = 3, {a,b} = 2, {{1,2,3},{4,5}} = 2 If S N, then we say Sis finite. Otherwise, we say S is infinite. What are some infinite sets we ve seen? Discrete Mathematics, Spring

13 Power Set Definition: Let Sbe a set. The power set (S) of a set Sis the set of all subsets of S. (S) = {x x S}. Example: ({a,b}) = {, {a}, {b}, {a,b}}. Sometimes (S) is written 2 S. Note that for finite S, (S) = 2 S. It turns out that P(N) > N. There are different sizes of infinite sets! Discrete Mathematics, Spring

14 Ordered n-tuples Definition: For n N, an ordered n-tupleor a sequenceoflength nis written (a 1, a 2,, a n ). The firstelement is a 1, etc. These are like sets, except that duplicates matter, and the order makes a difference. Note (1, 2) (2, 1) (2, 1, 1). Empty sequence, singlets, pairs, triples, quadruples, quintuples,, n-tuples. Discrete Mathematics, Spring

15 Cartesian Products of Sets Definition: Let Aand Bbe any two sets. The Cartesian product A B is defined to be A B ={(a, b) a A b B}. Example: {a,b} {1,2} = {(a,1),(a,2),(b,1),(b,2)} Note that for finite A, B, A B = A B. Note that the Cartesian product is not commutative: A B B A. Extends to A 1 A 2 A n... Discrete Mathematics, Spring

16 Union Operator Definition: Let A and B be any two sets. The niona Bof A and Bis the set containing all elements that are either in Aor( )in B(or, of course, in both), i.e., A B= {x x A x B}. Note that A B contains all the elements of Aandit contains all the elements of B:(A B A) (A B B) Discrete Mathematics, Spring

17 Example of Union {a,b,c} {2,3} = {a,b,c,2,3} {2,3,5} {3,5,7}= {2,3,5,3,5,7} ={2,3,5,7} Discrete Mathematics, Spring

18 Intersection Operator Definition: Let A and B be any two sets. The intersectiona Bof A and Bis the set containing all elements that are simultaneously in A and( ) in B, i.e., A B {x x A x B}. Note that A B is a subset of Aandit is a subset of B: (A B A) (A B B) Discrete Mathematics, Spring

19 Example of Intersection {a,b,c} {2,3} = {2,4,6} {3,4,5} = {4} Discrete Mathematics, Spring

20 Disjointedness Definition: Let Aand Bbe any two sets. A and Bare called disjoint(i.e., unjoined) iff their intersection is empty (A B= ). Example: The set of even integers is disjoint with the set of odd integers. Discrete Mathematics, Spring

21 Inclusion-Exclusion Principle How many elements are in A B? A B = A + B A B Example: How many students are on our class list? Consider set E =I M, I= {s sturned in an information sheet} M= {s s sent the TAs their address} Some students did both! E = I M = I + M I M Discrete Mathematics, Spring

22 Set Difference Definition: Let Aand Bbe any two sets. The set differenceof A and B, A Bis the set of all elements that are in Abut not B. A B = {x x A x B} = {x (x A x B ) } A Bis also called the complementofbwith respect to A. Discrete Mathematics, Spring

23 Example of Set Difference {1,2,3,4,5,6} {2,3,5,7,9,11} = {1,4,6} Z N ={, -1, 0, 1, 2, } {0, 1, } = {x xis an integer but not a nat. #} = {x xis a negative integer} = {,-3,-2,-1} Discrete Mathematics, Spring

24 Universal Set & Complement of a Set Definition(Universal Set): A set is a universal set orauniverse of discourse, denoted by U, if it includes every set under discussion. Definition(Complement of a Set): Let Abe a set. The complementof A inu, denoted by the set of all elements of U which are not elements of A, i.e., A = U A. U A= {x x U x A} A, is Example: If U=N, { 3,5} = {1,2,4,6,7,...} Discrete Mathematics, Spring

25 More on Set Complements An equivalent definition, when U is clear: A = { x x A} U A A Discrete Mathematics, Spring

26 Set Identities (Theorem) Theorem: Identity: A =A A U=A Domination: A U=U A = Idempotent: A A= A =A A Double complement: ( A ) = Commutative: A B=B A A B=B A Associative: A (B C)=(A B) C A (B C)=(A B) C A Discrete Mathematics, Spring

27 DeMorgan slaw for Sets Theorem: Exactly analogous to (and derivable from) DeMorgan s Law for propositions. A B = A B A B = A B Discrete Mathematics, Spring

28 Proving Set Equality To prove statements about the form E 1 = E 2 where E 1 and E 2 are sets, prove E 1 E 2 ande 2 E 1 separately. Discrete Mathematics, Spring

29 Example: Show A (B C)=(A B) (A C) Show A (B C) (A B) (A C). Assume x A (B C), & show x (A B) (A C). We know that x A, and either x Bor x C. Case 1: x B. Then x A B, so x (A B) (A C). Case 2: x C. Then x A C, so x (A B) (A C). Therefore, x (A B) (A C). Therefore, A (B C) (A B) (A C). Show (A B) (A C) A (B C). Discrete Mathematics, Spring

30 Theorem: If Aand Bare two sets, the following statements are equivalent. (1) A B (2) A B = A (3) A B = B Discrete Mathematics, Spring

31 Generalized Unions & Intersections Since union & intersection are commutative and associative, we can extend them from operating on ordered pairsof sets (A,B) to operating on sequences of sets (A 1,,A n ), or even unordered sets of sets, X={A Q(A)}. Discrete Mathematics, Spring

32 Generalized Union Binary union operator: A B n-ary union: A 1 A 2 A n : (( ((A 1 A 2 ) ) A n ) (grouping& order is irrelevant) Big U notation: n Ui =1 A i Or for infinite sets of sets: U A X A Discrete Mathematics, Spring

33 Generalized Intersection Binary intersection operator: A B n-ary intersection: A A 2 A n (( ((A 1 A 2 ) ) A n ) (grouping& order is irrelevant) Big Arch notation: I n i=1 A i Or for infinite sets of sets: I A X A Discrete Mathematics, Spring

34 Exercise 1.Let Aand Bbe sets. Show that (a) (A B) A (b)a (B-A) = A B 2.Let A, Band Cbe sets. Show that (A-B)-C= (A-C)-(B-C). 3.Let Aand Bbe two sets. Prove or disprove each of the followings (a) P(A) P(B) P(A B) where P(A) is the power set of the set A. (b) P(A B) P(A) P(B) Discrete Mathematics, Spring