@John #17 – that looks basically right to me, just three small typos:

1. "The inverse image is also called the preimage, and it's often written as \$$f^{-1}(X)\$$" – I think this should be \$$f^{-1}(S)\$$
2. "The image is often written as \$$f(X)\$$" – I think this should be \$$f(S)\$$
3. "\$$\\{y \in Y:\textrm{ for some } x \in T \\}\$$" – there's a missing \$$y = f(x)\$$