Let $f$ be a function from $X$ to $Y$ .For $B'\subset Y$, prove that $f(f^{-1}(B')) \subset B'$.

Question

MY ATTEMPT

Let us start by noticing that \begin{align*} f^{-1}(B') = \{x\in X \mid f(x)\in B'\} \Longrightarrow f(f^{-1}(B')) = \{f(x)\in Y \mid (x\in X)\wedge(f(x)\in B')\} \end{align*}

Then I get stuck. Could someone help me out?

Answer 1

By reading the question, I assume that $f$ is a function from $X$ to $Y$, and B' is a subset of Y (Please mention the hypothesis in your questions!). Let $y \in f(f^{-1}(B'))$. Then there exists $x \in f^{-1}(B')$ such that $y=f(x)$. But $x \in f^{-1}(B')$ so $f(x) \in B'$ and hence $y=f(x) \in B'$. So $f(f^{-1}(B'))\subseteq B'$.