Discrete Mathematics with Graph Theory (3rd Edition) 75

Discrete Mathematics with Graph Theory (3rd Edition) 75 -...

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Chapter 3 73 (b) 1- 1 can be defined on the range of I,that is, R '- {2}. (c) Let y = 1- 1 (x), x -=I- 2~ Then x = f(y) = ~. This implies y(2x - 4) = x. Since x -=I- 2, 2y -1 x ' x we have y = 2x _ 4' Thus 1- 1 (x) = 2x _ 4; 7. (a) The values are 7,5,3,1,2,4,6. (b) Let n E N. We must show that n = I(a) for some integer a. If n is even, let a = ~n. If n is odd, let a = -~(n - 1). ' 8. A function I on a set A is a reflexive relation if and only if I = i is the identity on A. Certainly, the identity function is reflexive. Conversely, if I is reflexive, then for any a E A, (a, a) E f. Since (a, x), (a, y) E I imply x = y, it follows ,that for any a E A, (a, x) E I if and only if x = a. Thus I=~ , 9. Lots of functions R -+ R are symmetric, for example, the function defined by I(x) = -x. 10. The greatest integer function is one example. For any a, b, e E R, if b = La J and e = L b J, then e = b because b is an integer, so e = La J . 11. (a) This is false:
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