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2.1 Basic set concepts. Section 2.1 Notes Page 1 A set is a collection of objects whose contents can be clearly determined. (Capital letters are generally used to name sets) The objects in a set are called the elements or members of the set. Example: Word Description; S is the set of states whose names begin with the letter A. Roster method; S = {Alabama, Alaska, Arizona, Arkansas} Set- Builder Notation; S = {x|x is a U.S. state whose name begins with the letter A} The empty set, also called the null set, is the set that contains no elements. The empty set is represented by { } or ∅. The symbol ∈ is used to indicate that an object is an element of a set. The symbol ∉ is used to indicate that an object is not an element of a set. The set of counting numbers is also called the set of Natural numbers and we represent this set by the bold face letter N. N = {1, 2, 3, …} The cardinal number of set A, represented by n(A), is the number of distinct elements in set A. Set A is equivalent to set B means that set A and set B contain the same number of elements. If set A and set B can be placed in a one to one correspondence, than A is equivalent to B: n(A) = n(B) Set A is a finite set if n(A) = 0 ( that is, A is the empty set) or n(A) is a natural number. S set whose cardinality is not 0 or a natural number is called an infinite set. Set A is equal to set B means that set A and set B contain exactly the same elements, regardless of order or possible repetition of elements. We symbolize the equality of sets A and B using the statement A = B. Ex1. Express each set using the roster method. a) Set C is the set of U.S. coins with a value of less than a dollar. b) Set M is the set of months beginning with the letter A. c) Set O is the set of a positive odd number less than 10. d) E = {X|X ∈ N and x is even} Ex2. Which one of the following is the empty set? a) {0} b) 0 c) {X|X is a number less than 4 or greater than 10} d) {X|X is a square with exactly three sides} e) {∅} Ex3. Determine whether each statement is true of false: a) r ∈ {a, b, c, ….,z} b) 7 ∉ {1, 2, 3, 4, 5} c) {a} ∈ {a, b} d) {Monday} ∈ {X|X is an day of the week} Ex4. Find the cardinal number of each of the following sets: a) A = {7, 9, 11, 13} b) B = {0} c) C = {13, 14, 15, …., 22, 23} d) ∅ e) D = {871} Section 2.1 Notes Page 2 Section 2.1 Notes Page 3 Ex5. Determine whether each statement is true or false: a) {4, 8, 9} = {8, 9, 4} b) {1, 3, 5} = {0, 1, 3, 5} c) {4, 5} = {5, 4, ∅}