I'm fairly new to math proofs. I've been looking for some counterexamples to the following theorems, especially the second one. I haven't been able to think of a scenario. Are the following theorems true?
- If $f: S \to T$ is an injection and $A \subseteq S$, then $f^{-1}(f(A))=A$.
- If $f: S \to T$ is a surjection and $C \subseteq T$, then $f(f^{-1}(C))=C$.
Making use of the injectivity of the function, We first prove the inclusion $f^{-1}(f(A)) \subset A$. let $x \in f^{-1}(f(A))$ then we have $f(x)\in f(A)$ which implies $x \in A$.
We now prove the inclusion $A \subset f^{-1}(f(A))$. let $x\in A$ then we have $f(x)\in f(A)$ which implies $x\in f^{-1}(f(A))$
A similar proof shows the second statement is also true.