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I am not sure about this question so I figured I would ask it on here. The question is: List all maps $\psi$ from $S = \{1,2\}$ to $T =\{-1,-2\}$ such that $\operatorname{Im}\psi = T$.

Is the question asking to make maps such that $\psi$ is a bijection? Would anyone be williing to give an explanation for this question or an example? Thank you!

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$S(1)=-1$ or $-2$. Because $\operatorname{Im}\psi=T$, $S(2)$ is determined now. Now it should be easy to give answers.

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If you have a surjective function $f$ between two finite sets with the same number $n$ of elements, this function must be bijective.

To show this, it suffices to show that it is injective. But if $f(x)=f(y)$, then the set of the images of $f$, say $\{f(a_1),\ldots,f(a_n)\}$ has less than $n$ elements, since there is at least an element that has been counted twice. This contradicts the surjectivity.

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