Similarly, is odd if the number of order reversals it performs is odd.
Example Let .
Thus is even because it performs 4 reversals, an even number. The property of evenness or oddness of a permutation is called its parity , by analogy with the evenness or oddness of integers. The parity of the product of two permutations is the sum their parities. The key to proving this is to notice that Indeed, if you first perform and then perform , then the total number of reversals is the number of reversals performed by plus the number reversals performed by , plus the number of cancellations that occur when a reversal performed by is itself reversed by . Each such cancellation cancels 2 reversals, one each from and , so reduces the total count by 2.