Final answer:
The sign function on permutations is unique for all permutations, as the sign can be determined by counting the number of transpositions required to obtain the permutation.
Step-by-step explanation:
The sign function on permutations is unique in the following sense:
- a. The sign function is not unique.
- b. The sign function is unique only for even permutations.
- c. The sign function is unique only for odd permutations.
- d. The sign function is unique for all permutations.
To prove that the sign function is unique for all permutations, we need to understand what the sign function is. The sign of a permutation is a measure of its parity, and it can be positive (+) or negative (-). An even permutation is one that can be obtained by an even number of swaps of elements, and an odd permutation is one that can be obtained by an odd number of swaps of elements.
To show that the sign function is unique for all permutations, we can consider the composition of two permutations. The sign of the composition is equal to the product of the signs of the individual permutations. Since any permutation can be decomposed into a series of transpositions, which are permutations that exchange two elements, we can see that the sign of any permutation is determined by the number of transpositions required to obtain it. If the number of transpositions is even, the sign is positive, and if the number of transpositions is odd, the sign is negative.
Therefore, the sign function is unique for all permutations, as the sign of a permutation can be determined by counting the number of transpositions required to obtain it.