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Theorem (The Composition Of Permutations Is A Permutation)
Let $X$ be a set. Suppose that $\sigma_1, \sigma_2, \ldots,\sigma_k$ are permutations of $X$. Then the composition $\sigma_1\circ \sigma_2\circ \ldots\circ \sigma_k$ is a permutation of $X$.
Problem 11 (The Composition Of Permutations Is A Permutation)
Prove theorem (The Composition Of Permutations Is A Permutation). As some reminders, you may use the following facts that you proved in either Math 301 or Math 340. You may use these facts without proof.
- The composition of two injective functions is injective.
- The composition of two surjective functions is surjective.
- A function is a bijection if it is both injective (1 to 1) and surjective (onto). Hence, the composition of two bijective functions is a bijection.
- You might find induction helps you get from a composition of two bijective functions is bijective to the composition of $n$ bijective functions is bijective.
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