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Discrete Mathematics with Graph Theory (3rd Edition) 61

Discrete Mathematics with Graph Theory (3rd Edition) 61 -...

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Section 3.2 n 1 2 3 m 1 1 2 3 2 1 4 9 30. [BB] 3 1 8 27 4 1 16 81 31. (a) LetA = {al,a2, ... ,a n }. 4 4 16 64 256 We guess that the number of functions X ---+ Y is n m. 59 If I: A ---+ B were one-to-one, then ICal), l(a2)' ... ' I(an) would be n different elements in B contradicting the fact that Bhas only m < n elements. (b) Let B = {bit b2, ... , bm}. If I: A -:-tB were onto, then there would exist elements alt a2, ... , am in A such that I(ai) == bi, distinct by~efinition of "function." But this contradicts the fact that A has n < m elements. 32. (a) [BB (--+)] Suppose A and B each contain n.elements. Assume that I: A ---+ B is one-to-one and let C = {I(a) I a E A}. Since leal) =/: 1(0.2) if al =/: a2, C is a subset of B containing n eiements; so C = B. Therefore, I is onto. Conversely, suppose I is onto. Then {I (a) I a E A} = B and so the set on the left here contains n distinct elements. Since Acontains only n elements, we cannot have I(al) = l(a2) for distinct ai, a2; thus, I is one-to-one. (b)
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