November 9, 2009

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Chapter 1 Sets

Deﬁnition 1.1. A set is a well-deﬁned collection of distinct objects. Each object in a set is called an element of the set. By “well-deﬁned”, we mean that the rule of membership to the set is clear. Example 1.2. The following are examples of sets. 1. The set of counting number less than 5. 2. The set of vowels in the word “mathematics”. 3. The set of cities in the Philippines. 4. The set of positive integers from −2 to 6, inclusive. 5. The set of days of the week. 6. The set of monkeys enrolled in Math 1. Objectives: 1. To deﬁne sets 2. To specify/ describe sets using the roster methods 3. To present the diﬀerent types of sets and the relationship between and among sets 4. To perform the basic operations on sets

Some Basic Notations

Capital letters such as A, B, C, are usually used to denote sets. Small letters such as a, b, c, are used to denote elements. If x is an element of set S, we write x ∈ S. If x is not an element of S, we write x ∈ S. /

Specifying Sets

The two basic methods of specifying a set are roster method and rule method. Deﬁnition 1.3. In the roster method, the elements of the set are listed, separated by commas, and enclosed in braces, { }. Example 1.4. Let A be the set of colors in the Philippine ﬂag. A roster form of A is A = {red, blue, yellow, white}. 3 Roster Method

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Rule Method

CHAPTER 1. SETS

Deﬁnition 1.5. In the rule method, a phrase describing precisely the elements is enclosed in braces. Example 1.6. Let A be the set of colors in the Philippine ﬂag. A rule form of A is A = {colors in the Philippine ﬂag}. An object x is an element of A provided x is a color in the Philippine ﬂag. Thus, another rule form of A is A = {x : x is a color in the Philippine ﬂag}. The symbol | or : means “such that”. More Example Write the given set using the roster and rule method. 1. A = The set of counting numbers less than 5. Roster Method: A = {1, 2, 3, 4}...

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