Sets and set operations

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1 CS 44 Discrete Mathematics for CS Lecture Sets and set operations Milos Hauskrecht 5329 Sennott Square Course administration Homework 3: Due today Homework 4: Due next week on Friday, February, 26 Midterm : Wednesday, February 5, 26 Covers chapter of the textbook Closed book Tables for equivalences and rules of inference will be given to you Course web page:

2 Review Definition: set is a (unordered) collection of objects. These objects are sometimes called elements or members of the set. (Cantor's naive definition) First seven prime numbers. X = { 2, 3, 5, 7,, 3, 7 } Definition: n ordered n-tuple (x, x2,..., xn) is the ordered collection that has x as its first element, x2 as its second element,..., and xn as its N-th element, N 2. Coordinates of a point in the 2-D plane (2, 6) Cartesian product Definition: Let S and T be sets. The Cartesian product of S and T, denoted by S x T, is the set of all ordered pairs (s,t), where s S and t T. Hence, S x T = { (s,t) s S t T}. Examples: S = {,2} and T = {a,b,c} S x T = { (,a), (,b), (,c), (2,a), (2,b), (2,c) } T x S = { (a,), (a, 2), (b,), (b,2), (c,), (c,2) } Note: S x T T x S!!!!

3 Cardinality of the Cartesian product S x T = S * T. = {John, Peter, Mike} ={Jane, nn, Laura} x = Cardinality of the Cartesian product S x T = S * T. = {John, Peter, Mike} ={Jane, nn, Laura} x = {(John, Jane),(John, nn), (John, Laura), (Peter, Jane), (Peter, nn), (Peter, Laura), (Mike, Jane), (Mike, nn), (Mike, Laura)} x =

4 Cardinality of the Cartesian product S x T = S * T. = {John, Peter, Mike} ={Jane, nn, Laura} x = {(John, Jane), (John, nn), (John, Laura), (Peter, Jane), (Peter, nn), (Peter, Laura), (Mike, Jane), (Mike, nn), (Mike, Laura)} x = 9 =3, =3 = 9 Definition: subset of the Cartesian product x is called a relation from the set to the set. Set operations Definition: Let and be sets. The union of and, denoted by, is the set that contains those elements that are either in or in, or in both. lternate: = { x x x }. = {,2,3,6} = { 2,4,6,9} =?

5 Set operations Definition: Let and be sets. The union of and, denoted by, is the set that contains those elements that are either in or in, or in both. lternate: = { x x x }. = {,2,3,6} = { 2,4,6,9} = {,2,3,4,6,9 } Set operations Definition: Let and be sets. The intersection of and, denoted by, is the set that contains those elements that are in both and. lternate: = { x x x }. = {,2,3,6} = { 2, 4, 6, 9} =?

6 Set operations Definition: Let and be sets. The intersection of and, denoted by, is the set that contains those elements that are in both and. lternate: = { x x x }. = {,2,3,6} = { 2, 4, 6, 9} = { 2, 6 } Disjoint sets Definition: Two sets are called disjoint if their intersection is empty. lternate: and are disjoint if and only if =. ={,2,3,6} ={4,7,8} re these disjoint?

7 Disjoint sets Definition: Two sets are called disjoint if their intersection is empty. lternate: and are disjoint if and only if =. ={,2,3,6} ={4,7,8} re these disjoint? Yes. = Cardinality of the set union Cardinality of the set union. = + - Why this formula?

8 Cardinality of the set union Cardinality of the set union. = + - Why this formula? Correct for an over-count. More general rule: The principle of inclusion and exclusion. Set difference Definition: Let and be sets. The difference of and, denoted by -, is the set containing those elements that are in but not in. The difference of and is also called the complement of with respect to. lternate: - = { x x x }. = {,2,3,5,7} = {,5,6,8} - =?

9 Set difference Definition: Let and be sets. The difference of and, denoted by -, is the set containing those elements that are in but not in. The difference of and is also called the complement of with respect to. lternate: - = { x x x }. = {,2,3,5,7} = {,5,6,8} - ={2,3,7} Complement of a set Definition: Let be the universal set: the set of all objects under the consideration. Definition: The complement of the set, denoted by, is the complement of with respect to. lternate: = { x x } ={,2,3,4,5,6,7,8} ={,3,5,7} =?

10 Complement of a set Definition: Let be the universal set: the set of all objects under the consideration. Definition: The complement of the set, denoted by, is the complement of with respect to. lternate: = { x x } ={,2,3,4,5,6,7,8} ={,3,5,7} = {2,4,6,8} Set identities Set Identities (analogous to logical equivalences) Identity = = Domination = = Idempotent = =

11 Set identities Double complement = Commutative = = ssociative ( C) = ( ) C ( C) = ( ) C Distributive ( C) = ( ) ( C) ( C) = ( ) ( C) DeMorgan ( ) = ( ) = bsorbtion Laws ( ) = ( ) = Complement Laws = = Set identities

12 Set identities Set identities can be proved using membership tables. List each combination of sets that an element can belong to. Then show that for each such a combination the element either belongs or does not belong to both sets in the identity. Prove: ( ) = _ _ Generalized unions and itersections Definition: The union of a collection of sets is the set that contains those elements that are members of at least one set in the collection. n i= i = { 2... } n Let i = {,2,...,i} i =,2,...,n n i = i = {, 2,..., n }

13 Generalized unions and intersections Definition: The intersection of a collection of sets is the set that contains those elements that are members of all sets in the collection. n I i = i = { 2... n } Let i = {,2,...,i} i =,2,...,n I n i = i = { } Computer representation of sets Idea: ssign a bit in a bit string to each element in the univesral set and set the bit to if the element is present otherwise use ll possible elements: ={ } ssume ={2,5} Computer representation: = ssume ={,5} Computer representation: =

14 = = Computer representation of sets The union is modeled with a bitwise or = The intersection is modeled with a bitwise and = The complement is modeled? = = Computer representation of sets The union is modeled with a bitwise or = The intersection is modeled with a bitwise and = The complement is modeled with a bitwise negation =

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