Question

Let B be a set of n elements. Prove that the number of different injective functions from B to B is n!. [n! was defined in Exercise 3.]

Short answer

It is proved the number of different injective functions from B to B isn!.

Step-by-step solution

Consider the given statement as follows

Suppose that the following property$P\left(n\right)$ as follows:

$2^n\le n!$

For all the integers,$n\ge 4$ .

Check the given inequality for n = 4.

First, prove the basic step for$n=4$ . Consider the following expression:

$$\begin{gathered} P\left(n\right)\le 2^n \\ {2}^4\le 4! \end{gathered}$$

So, the above inequality is true.

Check the given inequality for an integer k.

Now, suppose this works for any integerk to prove the inductive step.

$k\ge 4$

such that

$z^k\le k!$

(by inductive hypothesis).

So, the statement has been proved.