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- Homework Statement
- If ##f: A\rightarrow B## and ##A_0\subset A## and ##B_0\subset B##. (a) show that ##A_0\subset f^{-1}(f(A_0))## and that equality holds if ##f## is injective. (b) show that ##f(f^{-1}(B_0))\subset B_0## and that equality holds if ##f## is surjective.

- Relevant Equations
- The definitions of injective and surjective. Also definitions of ##f## and ##f^{-1}## restricted to a subset of the domain or range. I think I understand this part on my own but will type these up below if you want me to, I’m on my phone and TeX is a pain.

Mostly I need to clear up a few basic things about functions and their inverses, the problem seems easy enough. Ok, so for (a) I have

$$f^{-1}(f(A_0))= \left\{ f^{-1}(f(a)) | a\in A_0\right\}$$

but here I’m not certain if ##f^{-1}## is allowed to be multi-valued or not, the text says that if ##f## is bijective then ##f^{-1}## exists and is also bijective. But it didn’t specifically say “if, and only if” just if. And clearly the problem stipulates the existence of the inverse of ##f## even if it is not even injective so I’m unclear as to the nature of the inverse function here? I hope that makes sense.

$$f^{-1}(f(A_0))= \left\{ f^{-1}(f(a)) | a\in A_0\right\}$$

but here I’m not certain if ##f^{-1}## is allowed to be multi-valued or not, the text says that if ##f## is bijective then ##f^{-1}## exists and is also bijective. But it didn’t specifically say “if, and only if” just if. And clearly the problem stipulates the existence of the inverse of ##f## even if it is not even injective so I’m unclear as to the nature of the inverse function here? I hope that makes sense.

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