**problem:**

Show that if \(\displaystyle f:A\rightarrow B\) and E,F are subsets of A, then

\(\displaystyle f(E\cup F)=f(E)\cup f(F)\) and \(\displaystyle f(E\cap F)\subseteq f(E) \cap f(F)\).

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attempt:

attempt:

First I try to show that \(\displaystyle f(E\cup F)\subset f(E) \cup f(F)\) and then that \(\displaystyle f(E) \cup f(F) \subset f(E\cup F)\).

\(\displaystyle y\in f(E\cup F) \Rightarrow f^{-1}(y)\in E\cup F \Rightarrow f^{-1}(y)\in E \; or \; f^{-1}(y)\in F\).

\(\displaystyle f(f^{-1}(y))=y\in f(E\cup F)\) and so \(\displaystyle f(E\cup F)\subset f(E) \cup f(F)\).

\(\displaystyle y\in f(E)\cup f(F) \Rightarrow f^{-1}(y)\in E or f^{-1}(y)\in F \Rightarrow f^{-1}(y)\in E\cup F\).

\(\displaystyle f(f^{-1}(y))=y \in f(E\cup F)\) and so \(\displaystyle f(E) \cup f(F) \subset f(E\cup F)\).

A friend of mine pointed out that I could show this in the following manner:

\(\displaystyle

\begin{aligned}

f(E\cup F) =&\; \{f(x): x\in E \; or \; x\in F\}\\

=&\; \{f(x): x\in E\} \cup \{f(x): x\in F\}\\

=&\; f(E) \cup f(F)

\end{aligned}

\)

\(\displaystyle y\in f(E\cap F) \Rightarrow f^{-1}(y)\in E\cap F \Rightarrow f^{-1}(y)\in E \; and \; f^{-1}(y)\in F\).

Then \(\displaystyle f(f^{-1}(y))\in f(E)\cap f(F) \Rightarrow f(E\cap F)\subset f(E) \cap f(F).\)

I do not know how to continue. If \(\displaystyle y\in f(E) \cap f(F)\), what then? Should I be doing all of this differently?

Thank you.