2.1 Set Concepts 47 Any sets that are equal must also be equivalent. Not all sets that are equivalent are equal, however. The sets { } = D a b c , , and { } = E apple, orange, pear are equivalent because both have the same cardinal number, 3. Because the elements differ, however, the sets are not equal. Example 11 Equal and Equivalent Sets Determine whether the pairs of sets are equal, equivalent, both, or neither. a) A B alpha, beta, gamma, delta delta, beta, gamma, alpha { } { } = = b) C D George, John, Paul, Ringo Peter, Ace, Gene, Paul { } { } = = c) E F a, b, c, d, e, , z 1, 2, 3, , 100 { } { } = … = … Solution a) Even though the order of the elements is different, sets A and B contain the same elements. Therefore, = A B. Also, since = = n A n B ( ) ( ) 4, sets A and B are equivalent. Thus, sets A and B are both equal and equivalent. b) Notice that sets C and D do not contain the same elements, therefore the sets are not equal. However, since = = n C n D ( ) ( ) 4, sets C and D are equivalent. c) The elements in set E are letters while the elements in set F are numbers. Thus, the elements in the two sets are different and the two sets are not equal. Also, = = n E n F ( ) 26 and ( ) 100. Since the cardinal number of the two sets is different, the two sets are not equivalent. Therefore, sets E and F are neither equal nor equivalent. ■ Now try Exercise 73 Two sets that are equivalent or have the same cardinality can be placed in one-toone correspondence. Set A and set B can be placed in one-to-one correspondence if every element of set A can be matched with exactly one element of set B and every element of set B can be matched with exactly one element of set A. For example, there is a one-to-one correspondence between the student names on a class list and the student identification numbers because we can match each student with a student identification number. Consider set S, states, and set C, state capitals. S C North Carolina, Georgia, South Carolina, Florida Columbia, Raleigh, Tallahassee, Atlanta { } { } = = Two different one-to-one correspondences for sets S and C follow. S C North Carolina, Georgia, South Carolina, Florida Columbia, Raleigh, Tallahassee, Atlanta { } { } = = S C North Carolina, Georgia, South Carolina, Florida Columbia, Raleigh, Tallahassee, Atlanta { } { } = = Other one-to-one correspondences between sets S and C are possible. Some sets do not contain any elements, such as the set of zebras that live in your house. Definition: Empty Set The set that contains no elements is called the empty set or null set and is symbolized by { } or ∅.
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