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Let $A$ and $B$ be non-empty subsets of $\mathbb{R}$ and let $C$ be a non-empty subset of $A$.

If $f:A \to B$ is a function prove that

$$ C\subseteq f^{-1} [f(C)].$$

DKal
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S.Dan
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  • duplicate http://math.stackexchange.com/q/608707/129458 http://math.stackexchange.com/q/488757/129458 – OBDA May 27 '14 at 10:36

2 Answers2

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Just plug in the definitions. Take some $c \in C$. You want to show $c \in f^{-1}(f(C))$. What does that mean (definition)?

PhoemueX
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Actually the result is true when $A$, $B$ and $C$ are any sets. $\mathbb R$ plays no role here (nor the condition on $C$ being non-empty).

To prove the statement $X\subset Y$ given any $x\in X$ you must prove that $x\in Y$. In this case given any $c\in C$ you must prove that $c \in f^{-1}[f(C)]$. What $c \in f^{-1}(X)$ means? Can you check that $f(c) \in f(C)$?

Emanuele Paolini
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