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# Decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem. If false, provide an example, illustration, or brief explanation of why the statement is false. There are the same number of permutations of {1, 2, 3, 4} of even parity as there are of odd parity.

Solution

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The statement is true. $\{1,2,3,4\}$ has $4!=24$ distinct permutations. By theorem 3.1.7 we see that interchanges of adjacent numbers produces alternating changes in the parity of the permutations. Then, for each even parity permutation, we obtain one adjacent odd parity one. therefore, we have 12 even parity permutations and 12 odd parity ones.

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