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74-hw4s

# 74-hw4s - MATH 74 HOMEWORK 4 SOLUTIONS 1 How many elements...

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MATH 74 HOMEWORK 4 SOLUTIONS 1. How many elements does each of the following sets have? (a) { 1 , 2 , 3 , 4 } Four elements: 1, 2, 3, and 4 (b) {{ 1 , 2 } , { 3 , 4 }} Two elements: { 1 , 2 } and { 3 , 4 } (c) { 1 , { 2 , 3 } , { 4 , { 5 , 6 }}} Three elements: 1, { 2 , 3 } , { 4 , { 5 , 6 }} (e) { n 2 : n N and 1 n 9 } Nine elements: 1, 4, 9, 16, 25, 36, 49, 64, and 81 (f) { n 2 - n : n N and 1 n 9 } Nine elements: 0, 2, 6, 12, 20, 30, 42, 56, and 72 (g) { n 2 - 10 n + 9 : n N and 1 n 9 } Five elements: 0, - 7, - 12, - 15, and - 16 Remark. The point of this exercise was twofold: The fact that the elements of a set A can themselves be sets does not enter into a count of the elements of A The fact that if f is some function, the set { f ( n ) : 1 n 9 } does not necessarily have nine elements. It will have fewer than nine elements if f assigns the same value to diﬀerent numbers between 1 and 9 . This is an important point. The mere statement

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74-hw4s - MATH 74 HOMEWORK 4 SOLUTIONS 1 How many elements...

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