Question

Denote by SS the set of integers {−44,−43,...,−38−44,−43,...,−38}. Count the number of BIJECTIVE functions φ:S→Sφ:S→S such that φ(x)≠xφ(x)≠x for all x∈S.

Answer 1

Answer:

5039

Step-by-step explanation:

Since a bijective function of a set onto itself is nothing but a permutation of S and  the set  

S={-44, -43, -42, -41, -40, -39, -38}

has 7 elements, there are 7! = 5040 possible bijective functions of S onto itself.

But there is only one such that φ(x) = x

Hence, there are 5040-1 = 5039 bijective functions of S onto itself such that φ(x)≠x.

Answer 2

There are 5039 bijective functions such that  $\phi(x)\neq x$.

Set of integers given are,

               $S= (-44,-43,-42,-41,-40,-39,-38)$

A function f: A → B is a bijective function if every element b ∈ B and every element a ∈ A, such that f(a) = b and the element b is the image of the element a, and the element a is the pre image of the element b.

So, total number of bijective function of given set is the, factorial of number of element present in given set.

Total number of element in given set is, = 7

Thus, the number of BIJECTIVE functions = $7!$

                                                                     = $7*6*5*4*3*2*1=5040$

In 5040 bijective functions, one element will be satisfy $\phi(x)=x$

So, number of bijective function such that $\phi(x)\neq x$ is,

                                                $=5040-1=5039$

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