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Discrete Mathematics with Graph Theory (3rd Edition) 72

# Discrete Mathematics with Graph Theory (3rd Edition) 72 -...

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70 Solutions to Exercises 20. (a) [BB] This set is uncountable. The function defined by f(x) = x-I gives a one-to-one corre- spondence between it and (0,1), which we showed in the text to be uncountable. (b) This set is countably infinite. Just follow the sequence given in the text for the set of all positive rationals, but omit any rational number not in (1,2). (c) This set is finite. In fact, it contains at most 99 2 elements since there are 99 possible numerators and, for each numerator, 99 possible denominators. (d) This set is countably infinite. List the elements as follows (deleting any repetitions such as 5/100 - 1/20)· 99 99 99 98 98 98 - ·6'7'···' 104'6'7'···'W4'··· . (e) [BB] This set is countably infinite. In Exercise 3(c) we showed it is in one-to-one correspondence with N x N. (f) This set is countably infinite because it is in one-to-one correspondence with Q via the function f defined by f(a, b) = a. (g) This set is uncountable. It is in one-to-one correspondence with [-1,1] = {x I -1 ~ x ~ I} via the function f defined by f(a, b) = a. The interval [-1,1]
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