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} Rule Method: A = {x | x is a counting number less than 5} = {counting numbers less than 5} 2. B = The set of vowels in the word “mathematics”. Roster Method: B = {a, e, i} Rule Method: B = {y | y is a vowel in the word “mathematics”} = {vowels in the word “mathematics”} 3. C = The set of all negative integers. Roster Method: C = {. . . , −3, −2, −1} Rule Method: C = {z | z is a negative integer} = {negative integers} 4. D = The set of positive integers from − 2 to 6, inclusive. Roster Method: D = {1, 2, 3, 4, 5} Rule Method: D = {x | x is a positive integer from − 2 to 6, inclusive} = {positive integers from − 2 to 6, inclusive} 5. E = The set of all counting numbers less than100. Roster Method: E = {1, 2, 3, . . . , 98, 99} Rule Method: E = {y | y is a counting number less than 100} = {counting numbers less than 100}

SOME BASIC SETS

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Some Basic Sets

Deﬁnition 1.7. A set without any element is an empty set or null set, and is denoted by { } or ∅. Deﬁnition 1.8. A set which has only one element is called a unit set. Deﬁnition 1.9. A set containing all the elements under consideration is called a universal set, and is denoted by U . Example 1.10. The following are universal sets of A = {a, i, o, u}, B = {a, b, c}, C = {w, s, t}: 1. U = {x | x is a letter in the English alphabet}. 2. U = {a, b, c, i, o, u, w, s, t} 3. U = {a, b, c, i, o, u, w, s, t, 1, 2} Null Set Unit Set Universal Set

Related Sets

Deﬁnition 1.11. Two sets A and B are said to be equal, and we write A = B, if each element of set A is an element of set B and each element of set B is an element of set A. Example 1.12. The sets A = {a, b, c} and B = {c, a, b} are equal. Deﬁnition 1.13. If it is possible to pair each element of a set A with exactly one element of a set B and each element of B with exactly one element of A, then we say that the sets A and B can be arrayed in a one-to-one correspondence. Example 1.14. 1. The sets A = {a, b, c} and B = {1, 2, 3} are in one-to-one correspondence....