Discrete Mathematics with Graph Theory (3rd Edition) 67

# Discrete Mathematics with Graph Theory (3rd Edition) 67 -...

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Section 3.2 65 25. [BB] Since a bijective function is, by definition, a one-to-one onto function, we conclude, by the results of part (a) of the previous two exercises, that indeed the composition of bijective functions is bijective. 26. (a) f(1000) = 998; f(999) = fU(1003)) = f(1001) = 999; f(998) = fU(1002)) = f(1000) = 998; f(997) = fU(1001)) = f(999) = 999. (b) We guess that f(n) =. . . { 998 if n is even 999 If n IS odd. (c) We guess rng f = {998,999}. 2 2 () () Xl X2 Xl X2 2 2 2 2 2 27. Suppose f Xl = f X2 . Then ~ = ~,so ~2 = ~2' XIX2 + 2XI = XIX2 + Y xi + 2 y x~ + 2 xl + x 2 + 2x~, 2xi = 2x2 and Xl = ±X2. Note that X2 = -Xl is not possible (unless Xl = X2 = 0) since -Xl Xl . f( -Xl) = ~ =I- ~ = f(XI)' Thus f IS one-to-one. Next we note that yxi + 2 yxi + 2 X Y E rng f ~ Y = f (x) for some X E R ~ Y = .y'X"2+2 for some X E R. X2 +2 Now, if y = ~, then y2(X2 + 2) = X2, so x2(y2 - 1) = _2y2 and this implies that y2 =I- 1 X2 +2 (otherwise we have the equation 0 = 1) and also y2 -'-1 :::;
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## This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.

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