Given a permutation on a set, show that its inverse is again a permutation. I started the problem: Let f be a permutation on a set A. Let g be the inverse of f. Show that g is a function, 1-1, and onto. I'm having trouble understanding exactly what it means for g to be a function. Do I need to show it is well defined? If so, then I know since f is 1-1 and onto that each element in A goes to exactly one other element in A. So if I'm "going backwards" when I think about g, then shouldn't g too be well defined?
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Yes, your intuition is completely correct. Whoever posed this problem probably just wants you to be able to write down your reasoning formally.
This all may seem like overkill and, yes, it is rather silly. Soon you'll be moving onto more interesting things. Have fun!