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Let $A, B$ be two subsets of a set $X$, and let $f : X \rightarrow Y$ be a function. Show that $f(A \cup B) \subseteq f(A) \cup f(B)$, $f(A) \setminus f(B) \subseteq f(A \setminus B)$. Is it true that the $\subseteq$ relation can be improved to $=$?
I proved $f(A \cup B) \subset f(A) \cup f(B)$ and $f(A) \setminus f(B) \subset f(A \setminus B)$. But i'm not sure when we improved the $\subseteq$ by $=$. Can anyone give me some examples when the relation is $=$?