3.3N_Sets_and_Algebraic_Structures

# 3.3N_Sets_and_Algebraic_Structures - Math 280 Strategies of...

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Unformatted text preview: Math 280 Strategies of Proof Fall 2007 Class Notes 3.3 Sets and Algebraic Structure Describing Sets Like the concepts of point and line in Euclidean geometry or the concepts of true or false in logic, the terms set and element are not formally defined in mathematics. While we cannot give a formal definition of a set, we do use the following informal definition: A set is a well-defined collection of objects. The objects are called the elements of the set. We use the notation x ∈ S to indicate that x is an element of the set S and x / ∈ S to denote that x is not an element of S . We typically use upper-case letters, such as A , B , or S , to denote sets and lower-case letters, such as a , b , x , or y , to denote elements or possible elements of sets. There are two general methods for specifying sets: the list method and the rule method . Depending upon the specific set, one method may be easier or more convenient to use than the other. In many cases, we must use one method over the other. Using the list method , we specify a set by explicitly listing its elements, separated by commas, enclosed in braces. For instance, A = { 1 , 2 , 3 } specifies a set containing three elements, 1, 2, and 3. A set is completely determined by its elements. The order in which the elements are listed does not change the set. Listing an element more than once also does not change the set. Therefore the following all specify the same set { a, b, c } , { b, c, a } , { a, a, c, b, b, b } . Since listing an element more than once does not change the set, we should list elements only once. The rule method describes a set by specifying one or more rules, or properties, which the elements must satisfy in order to belong to the set. The set is written using set-builder notation as in S = { x ∈ D | p ( x ) } , where p ( x ) is a propositional function specifying the properties that elements of the set must satisfy. The vertical line in this notation is read “such that” and the the notation is read “the set of all x in D such that p ( x ) is true.” An element in the domain D belongs to S provided that the propositional function p ( x ) is true for that element. Example. The following set is specified using the rule method: A = { x is an integer | − 1 ≤ x < 5 } Identify the domain of x , the rule, and write the set using the list method. Solution. The domain is the set of integers and the rule is the propositional function p ( x ) : − 1 ≤ x < 5 . Using the list method, we can write this set as A = {− 1 , , 1 , 2 , 3 , 4 } . Example. The following set is specified using the rule method: B = { x is a real number | x 2 − 2 x + 1 = 0 } Identify the domain of x , the rule, and write the set using the list method....
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3.3N_Sets_and_Algebraic_Structures - Math 280 Strategies of...

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