i wanted to prove:
Let $f:X\to Y$ be a mapping from $X$ into $Y$. Show that if $A$ and $B$ are subsets of $X$, then:
$$(A \subset B) \implies \left(f(A) \subset f(B)\right)$$
but i thought about it and it isnt right. if it were $(A \subset B) \implies \left(f(A) \subseteq f(B)\right)$ i would believe it, cause we can take the function $f$ to make all elements from $X$ go to $1$ in $Y$, and this means that i can have $A=\{2\}, B=\mathbb{Z}$ and clearly $A\subset B$, but $f(A)=f(B)$ e.g. $f(A)\subseteq f(B)$ not what they say.
what is wrong?
If you need $f(A)$ to be a proper subset of $f(B)$, then you're right, it's not true. Your counterexample is great. Many times, people use the symbol $\subset$ to mean $\subseteq$, and use $\subsetneqq$ if they want to indicate a proper subset. Perhaps that's what's going on with this exercise?