1. The inverse function. It makes sense that it needs to be both injective and surjective. A function has to have every value of the set A as the first coordinate, so naturally if not every value of the set B is in the range of the function, then when you take the inverse and switch the coordinates, then it wouldn't be a function anymore. So it has to be surjective. And it has to be one-to-one because all the points have to be unique and distinct, whether it's (a,b) or (b,a). The book shows a few examples that are easy to find by algebraic manipulation...so what do we do if it's not possible to find the inverse by manipulation? Is there a way? I don't know if we're responsible to know how because the book didn't teach us how, but it would still be interesting to know.
2. So a permutation...I feel like I've heard about these before. I forgot what they are. So a permutation is just a function of A-->A that is one to one? So basically a subset. That sounds good, but the use of it is not totally known to me. I seem to have this problem all the time. I know how to calculate something, but don't know why I would want to do it. Other than the format that we can write it in, I don't quite know what the difference between permutations and everything else we've been looking at is.
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