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I can't figure out why we need $\phi$ to be bijective in order to conclude $|A_n| = |S_n| - |A_n|$, could someone please explain how this works?

Warz
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    If you have a bijection between two sets $X,Y$, you know the two sets have equally many elements. If you have just any map, you cannot conclude anything about the sizes. – Daniel Fischer Apr 01 '14 at 20:33

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