46 CHAPTER 2 Sets Another important set concept is the equality of sets. Solution a) Set A has 6 elements. Since 6 is a natural number, set A is a finite set. b) The ellipsis indicates that Set B continues forever. So, the number of elements in set B is not a natural number. Therefore, set B is an infinite set. c) Set C is a very large set. But, set C has 1,000,000 elements and 1,000,000 is a natural number. Therefore, set C is a finite set. d) Set D contains fractions including , , , , , , , . 1 2 1 3 1 4 1 5 1 6 1 7 1 8 … Set D contains many other fractions as well. Thus, the number of elements in set D is not a natural number and set D is an infinite set. ■ Now try Exercise 19 Definition: Equal Sets Set A is equal to set B, symbolized by = A B, if and only if set A and set B contain exactly the same elements. For example, if set { } = A 1, 2, 3 and set { } = B 3, 1, 2 , then = A B because they contain exactly the same elements. The order of the elements in the set is not important. If two sets are equal, both must contain the same number of elements. The number of elements in a set is called its cardinal number. Definition: Cardinal Number The cardinal number of set A, symbolized by n A( ), is the number of elements in set A. Both set { } = A 1, 2, 3 and set { } = B England, Brazil, Japan have a cardinal number of 3; that is, = n A( ) 3, and = n B( ) 3. We can say that set A and set B both have a cardinality of 3. Example 10 Cardinal Number of Sets Use the sets A B cat, dog, rabbit, parakeet, goldfish, iguana , 1, 2, 3, , 41 , { } { } = = … C and a, b, c, , z . { } = … Determine a) n A( ) b) n B( ) c) n C( ) Solution a) Set A has 6 elements, therefore = n A( ) 6. b) Set B has 41 elements, therefore = n B( ) 41. c) Set Chas 26 elements, therefore n A( ) 26. = ■ Now try Exercise 65 Two sets are said to be equivalent if they contain the same number of elements. Definition: Equivalent Sets Set A is equivalent to set B if and only if = n A n B ( ) ( ).
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