Prove that : if $L \subseteq M,$ then $f^{-1}(L) \subseteq f^{-1}(M)$.
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Here is how you advance
$$ x \in f^{-1}(L)\implies f(x) \in L \implies f(x)\in M \implies x\in f^{-1}(M) \implies L \subset M.$$
You should know how to justify the steps in the proof.
Mhenni Benghorbal
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The basic property of $\;\cdot^{-1}[\cdot]\;$ is $$ x \in f^{-1}[Y] \;\equiv\; f(x) \in Y $$ So let's take our proof goal, use the definitions and this basic property, and see where that leads us: \begin{align} & f^{-1}[L] \;\subseteq\; f^{-1}[M] \\ \equiv & \;\;\;\;\;\text{"definition of $\;\subseteq\;$"} \\ & \langle \forall x :: x \in f^{-1}[L] \;\Rightarrow\; x \in f^{-1}[M] \rangle \\ \equiv & \;\;\;\;\;\text{"the above basic property, twice"} \\ & \langle \forall x :: f(x) \in L \;\Rightarrow\; f(x) \in M \rangle \\ \equiv & \;\;\;\;\;\text{"..."} \\ \end{align} Now use the assumption $\;L \subseteq M\;$, or in other words $\;\langle \forall y :: y \in L \;\Rightarrow\; y \in M \rangle\;$, and complete the proof.
