©1999 Indiana University Trustees Basic Set Theory Definitions A set is a collection of objects or elements An element is an object that make up a set The universal set (parent set) contains ALL of the elements considered in a given problem; serves as an encompassing set for various subsets. (contains various subsets)

©1999 Indiana University Trustees Simple Set Example the universal set is a deck of ordinary playing cards each card is an element in the universal set some subsets are: –face cards –numbered cards –suits –poker hands

©1999 Indiana University Trustees Universal Sets The universal set is the set of all things pertinent to to a given discussion and is designated by the symbol U Example: U = {all students at IUPUI} Some Subsets: A = {all Computer Technology students} B = {freshmen students} C = {sophomore students}

©1999 Indiana University Trustees Subsets a subset whose elements are all members of another set notation:  means “is a subset of”  means “is a proper subset of”  means “is not a subset of”

©1999 Indiana University Trustees The Empty Set empty set (null set) - any set that contains no elements (the empty set is a subset of every set including itself)

Union - the set of all the elements in two or more given sets. Intersection – the set of all the elements that are common to two or more given sets. Disjoint set – a set with no elements in common. Complement – the set of all elements in the universal set that are not in a given set, which is a subset of the universal set. ©1999 Indiana University Trustees

Venn Diagrams Venn diagrams show relationships between sets and their elements Universal Set Sets A & B

©1999 Indiana University Trustees Venn Diagram Example 1 Set DefinitionElements A = {x | x  Z + and x  8} B = {x | x  Z + ; x is even and  10} A  B B  A

©1999 Indiana University Trustees Venn Diagram Example 2 Set DefinitionElements A = {x | x  Z + and x  9} B = {x | x  Z + ; x is even and  8} A  B B  A A  B

©1999 Indiana University Trustees Venn Diagram Example 3 Set DefinitionElements A = {x | x  Z + ; x is even and  10} B = x  Z + ; x is odd and x  10 } A  B B  A

©1999 Indiana University Trustees Venn Diagram Example 4 Set Definition U = {1, 2, 3, 4, 5, 6, 7, 8} A = {1, 2, 6, 7} B = {2, 3, 4, 7} C = {4, 5, 6, 7} A = {1, 2, 6, 7}

©1999 Indiana University Trustees Venn Diagram Example 5 Set Definition U = {1, 2, 3, 4, 5, 6, 7, 8} A = {1, 2, 6, 7} B = {2, 3, 4, 7} C = {4, 5, 6, 7} B = {2, 3, 4, 7}

©1999 Indiana University Trustees Venn Diagram Example 6 Set Definition U = {1, 2, 3, 4, 5, 6, 7, 8} A = {1, 2, 6, 7} B = {2, 3, 4, 7} C = {4, 5, 6, 7}