Mapping/Take image/Exercise

Let ${}F\colon L\rightarrow M$ be a mapping. Show that taking images

${\mathfrak {P}}\,(L)\longrightarrow {\mathfrak {P}}\,(M),S\longmapsto F(S),$

satisfies the following properties (for arbitrary subsets ${}S,S_{1},S_{2}\subseteq L$):

1. ${}F(S_{1}\cap S_{2})\subseteq F(S_{1})\cap F(S_{2})$,
2. ${}F(S_{1}\cup S_{2})=F(S_{1})\cup F(S_{2})$,
3. ${}F(L\setminus S)\supseteq F(L)\setminus F(S)$.

Show by examples that the inclusions in (1) and (3) might be strict.

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