To show that the number of all even permutations is the same as the number of all odd permutations, one of the following is true: For any map we define, the well-defined property need not to be proven. If there is a map defined from the set of all even permutations to the set of all odd permutations, we show it is onto by choosing a permutation of the form Oh where o is an odd permutation, and M is an odd permutation. Define a map from the set of all even permutations to the set of all odd permutations by (u) = (23) For any map we define we need to show that it is just a homomorphism