Question

Question 24: Let ${\mathbf{B}}_{\mathbf{n}}$ denote the set of odd permutations in ${\mathit{S}}_{\mathbf{n}}$. Define a function$\mathbf{f}\mathbf{:}{\mathbf{A}}_{\mathbf{n}}\to {\mathbf{B}}_{\mathbf{n}}$by$\mathbf{f}\left(\mathbf{\alpha }\right)=\left(\mathbf{12}\right)\mathbf{\alpha }$ .

1. Prove that f is injective.
2. Prove that f is surjective. [Hint: If $\mathbf{\beta }\in {\mathbf{B}}_{\mathbf{n}}$, then ${\mathit{A}}_{\mathbf{n}\mathbf{}}\left(12\right)\mathbf{\beta }\mathbf{\in }{\mathbf{A}}_{\mathbf{n}}$.] So f is bijective. Hence, ${\mathit{A}}_{\mathbf{n}}$and ${\mathit{B}}_{\mathbf{n}}$have the same number of elements.

Show that $\left|A_n\right|\mathbf{=}\frac{\mathbf{n}\mathbf{!}}{\mathbf{2}}$. [Hint: Every element of ${\mathit{S}}_{\mathbf{n}}$is in ${\mathit{A}}_{\mathbf{n}}$or${\mathit{B}}_{\mathbf{n}}$ (but not both) and $\left|{\mathbf{S}}_{\mathbf{n}}\right|=\mathbf{n}\mathbf{!}$.]

See Exercise 39(a) and (b) for a generalization of this exercise.

Short answer

b) It is proved that f is surjective

Step-by-step solution

Step-by-Step Solution Step 1: Injective function

A surjective function is a function if every contains at least one pre- image $x\in domain$such that $f\left(x\right)=y$.

Show that is surjective

b)

With$\beta \in B_n$, there is $\left(12\right)\beta \in A_n$(Since $\beta$has an odd number of transpositions and since $\left(12\right)\beta$would be transpositions then $\left(12\right)\beta$, has an even number of transpositions) and therefore,

$$\begin{gathered} f\left(\left(12\right)\beta \right)=\left(\left(12\right)\left(12\right)\beta \right) \\ =\left(\left(1\right)\beta \right) \\ =\beta \end{gathered}$$

As a result ,$f$is surjective. Combined with (a), this implies that is bijective and therefore $\left|A_n\right|=\left|B_n\right|$

Hence, it is proved thatf is surjective