Let $A$ and $B$ sets, with $P,Q \subseteq A$ and let $f:A \to B$

1) prove that $f(P)-f(Q) \subseteq f(P-Q)$

2)Is it necessarily the case that $f(P-Q) \subseteq f(P)-f(Q)$? Give a proof or a counterexample

How I can to proceed?


(1) Is quite straight forward: Let $b \in f[P] - f[Q]$.Then $b \in f[P]$, $b \not\in f[Q]$. By definition of $f[P]$, there is an $p \in P$ with $b = f(p)$. As $b \not\in f[Q]$, $p \not\in Q$. Hence $p \in P - Q$ and therefore $b \in f[P-Q]$.

Hint for (2). Think of constant functions, let for example $A = B = \mathbf R$ and let $f$ be a constant functions.


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