Chapter 1.2, Problem 25E

Let $f:A\to B$, where $A$ and $B$ are non empty, and let $T_1$ and $T_2$ be subsets of $B$.

Prove that $f^{-1}\left(T_1\cup T_2\right)=f^{-1}\left(T_1\right)\cup f^{-1}\left(T_2\right)$.

Prove that $f^{-1}\left(T_1\cap T_2\right)=f^{-1}\left(T_1\right)\cap f^{-1}\left(T_2\right)$.

Prove that $f^{-1}\left(T_1\right)-f^{-1}\left(T_2\right)=f^{-1}\left(T_1-T_2\right)$.

Prove that if $T_1\subseteq T_2,then{f}^{-1}\left(T_1\right)\subseteq f^{-1}\left(T_2\right)$.