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### Proof Regarding an iff Statement [duplicate]

Let $f:X \to Y$ be a function. Then $f$ is one-to-one iff for all subsets $A$ and $B$ of $X$, $f(A\cap B) = f(A) \cap f(B)$. Any proofs or guidance would be greatly appreciated
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### Do we have always $f(A \cap B) = f(A) \cap f(B)$? [closed]

Suppose $A$ and $B$ are subsets of a topological space and $f$ is any function from $X$ to another topological space $Y$. Do we have always $f(A \cap B) = f(A) \cap f(B)$? Thanks in advance
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### Proving $f(C) \setminus f(D) \subseteq f(C \setminus D)$ and disproving equality

Let $f: A\longrightarrow B$ be a function. 1)Prove that for any two sets, $C,D\subseteq A$ , we have $f(C) \setminus f(D)\subseteq f(C\setminus D)$. 2)Give an example of a function $f$, and sets $C$...
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### Prove $F(F^{-1}(B)) = B$ for onto function

Suppose that $f:X \to Y$ is an onto function. Prove that for all subsets $B$ subset of $Y$, $f(f^{-1}(B)) = B$. I don't know how to do this if the function is not also one to one, which it is not. Any ...
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### Why $f^{-1}(f(A)) \not= A$ [duplicate]

Let $A$ be a subset of the domain of a function $f$. Why $f^{-1}(f(A)) \not= A$. I was not able to find a function $f$ which satisfies the above equation. Can you give an example or hint. I was asking ...
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### how to prove $f^{-1}(B_1 \cap B_2) = f^{-1}(B_1) \cap f^{-1}(B_2)$

I am given this equation: $f^{-1}(B_1 \cap B_2) = f^{-1}(B_1) \cap f^{-1}(B_2)$ I want to prove it: what i did is I take any $a \in f^{-1}(B_1 \cap B_2)$, then there is $b \in (B_1 \cap B_2)$ so ...
### $f(A \cap B)\subset f(A)\cap f(B)$, and otherwise?
I got a serious doubt ahead the question Be $f:X\longrightarrow Y$ a function. If $A,B\subset X$, show that $f(A \cap B)\subset f(A)\cap f(B)$ I did as follows \forall\;y\in f(A\cap B)\...