Odd and Even Permutations of Set are Equivalent

Theorem
Let $n \in \N_{> 0}$ be a natural number greater than $0$.

Let $S_n$ denote the symmetric group on $n$ letters.

Let $R_e$ and $R_o$ denote the subsets of $S_n$ consisting of even permutations and odd permutations respectively.

Then $R_e$ and $R_o$ are equivalent.

Proof
Let $\tau$ be a transposition in $R_o$.

Then any element of $R_o$ can be expressed in the form:


 * $\rho \circ \tau$

for some $\rho \in R_e$.

Thus a bijection $\phi$ can be constructed:
 * $\forall \rho \in R_e: \phi \left({\sigma}\right) = \rho \circ \tau$

Hence the result by definition of set equivalence.