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MTH 231 Section 2.1 Sets and Operations on Sets

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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.

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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.

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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”}

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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.

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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.

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Pictures

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Complement The complement of a set A is all the elements in the universal set U that are not elements in set A.

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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 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).

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Empty Set The empty set is a set with no elements.

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Intersection The intersection of two sets A and B is the set of elements common to both A and B.

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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).

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Union The union of two sets A and B is the set of elements that are in A or B (or both).

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An Example p. 75 #9

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