Determine which of the following functions g : z - z are permutations of z (a) g is defined by g(a) = -a + 2 (b) g is defined by g(a) = a^3.

(a) It is a permutation of Z

(b) It is not a permutation of Z

Step-by-step explanation:

Recall that a permutation of a set S, is just a function g:S--->S which is both injective and surjective. That is to say, g is bijection from S onto S.

(a)

Let's show that g is injective, to do this, we need to show

But  

So g is injective.

To prove g is surjective, let's take any integer b and show that there exists an integer a such that g(a) = b.

If we take the integer a = 2-b, then

g(a) = g(2-b) = -(2-b)+2 = -2 +b +2 = b.

So g is also surjective, and thus, a permutation of the integers.

(b)

Although g is injective, g fails to be surjective. To show this, let's show that there is no integer a such that g(a) = 2.

but a is not an integer, and g is not surjective. So it is not a permutation.