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MTH 231 Section 2.1 Sets and Operations on Sets Overview • The notion of a set (a collection of objects) is introduced in this chapter as the primary way to describe whole numbers. • Operations (e.g., union and intersection) on sets form the basis for addition, subtraction, multiplication, and division. • We will explore different models of these operations, and subsequently properties of whole numbers through these models. Sets • A set is a collection of objects. • An object that belongs to a particular set is called an element, or member. • Sets must be well-defined: 1. there must be a universe of objects that are allowed into consideration; 2. each object either is or is not an element of the set. Three Ways To Define A Set 1. Word Description. The letters in the word “alabama” 2. Listing in Braces. {a, l, b, m} 3. Set-builder Notation. {x | x is one of the letters in the word “alabama”} More • The order in which elements are listed is arbitrary. • Elements should be listed just once. • Capital letters are generally used to denote, or name, sets. • Membership in a set is represented by ϵ. Venn Diagrams • A pictorial representation of sets. • The universal set, denoted by U, is represented by a rectangle. • Any sets under discussion are represented by loops inside the rectangle. • The region inside the loop is associated with the elements of the set. Pictures Complement • The complement of a set A is all the elements in the universal set U that are not elements in set A. Subset • The set A is a subset of another set B if, and only if every element of A is also an element of B. A B • A is a proper subset of B if A is a subset of B but A and B are not equal (two sets are equal if they have precisely the same elements). Empty Set • The empty set is a set with no elements. Intersection • The intersection of two sets A and B is the set of elements common to both A and B. A B Disjoint Sets • Two sets A and B are disjoint if A and B have no elements in common (or, that the intersection of A and B is the empty set). A B Union • The union of two sets A and B is the set of elements that are in A or B (or both). A B An Example • p. 75 #9