Discrete Structures – CNS 2300 Text Discrete Mathematics and Its Applications (5th Edition) Kenneth H. Rosen Chapter 1 The Foundations: Logic, Sets, and Functions

Section 1.7 Set Operations

Unions of Sets Let A and B be sets. The union of the sets A and B, denoted by , is the set that contains those elements that are either in A or in B, or in both. union A B

Unions of Sets Let A and B be sets. The union of the sets A and B, denoted by , is the set that contains those elements that are either in A or in B, or in both. A = {2, 4, 6, 8, 10} B = {1, 2, 3, 4, 5} ={1, 2, 3, 4, 5, 6, 8, 10}

Intersection of Sets Let A and B be sets. The intersection of the sets A and B, denoted by , is the set containing those elements in both A and B. intersection A B

Intersection of Sets Let A and B be sets. The intersection of the sets A and B, denoted by , is the set containing those elements in both A and B. A = {2, 4, 6, 8, 10} B = {1, 2, 3, 4, 5} ={2, 4}

Disjoint Sets Two sets are called disjoint if their intersection is the empty set. intersection empty A B

Disjoint Sets Two sets are called disjoint if their intersection is the empty set. A = {4, 6, 8, 10} B = {1, 2, 3, 5} ={ }

Difference of Sets Let A and B be sets. The difference of A and B, denoted by A-B, is the set containing those elements that are in A but not in B The difference of A and B is also called the complement of B with respect to A. difference A B

Difference of Sets Let A and B be sets. The difference of A and B, denoted by A-B, is the set containing those elements that are in A but not in B The difference of A and B is also called the complement of B with respect to A. A = {2, 4, 6, 8, 10} B = {1, 2, 3, 4, 5} A-B ={6, 8, 10}

Complement of a Set Let U be the universal set. The complement of the set A, denoted by A, is the complement of A with respect to U. In other words, the complement of the set A is U - A. A U complement

Complement of a Set Let U be the universal set. The complement of the set A, denoted by A, is the complement of A with respect to U. In other words, the complement of the set A is U - A.

Set Identities x*1 = x Identity laws x+0 = x Domination laws x*0 = 0 Idempotent laws -(-x)= x Double negation law

Set Identities x+y = y+x Commutative laws x*y = y*x Associative laws Distributive laws DeMorgan’s laws

Set Identities

Proving Equivalences Use set builder notation and logical equivalences. Use membership tables.

Generalized Unions and Intersections 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. The intersection of a collection of sets is the set that contains those elements that are members of all the sets in the collection.

Computer Representation of Sets Bit vector representation Representations that contain the actual elements of the set.

Problems from the text Homework will not be collected. However, you should do enough problems to feel comfortable with the concepts. For these sections the following problems are suggested. Pages 94-96 1-21 odd 41, 43

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