Let $X,Y$ two finite sets with same cardinality and let $f:X \rightarrow Y$ an injective map. Then $f$ is surjective.
I'm sketching down how I'm trying to solve this exercise and I would like to know if it's very far from being correct or is it an acceptable solution. I'm trying by contradiction:
suppose ther exists some $\bar{y} \in Y$ such that $\forall x \in X \, , \, \phi(x) \neq \bar{y}$
But this implies that there exist at least one point $y^* \in Y$ such that $\phi(x_1)=y^*=\phi(x_2)$ for some $x_1,x_2 \in X$, with $x_1 \neq x_2$ but this would contradict the injectivity hypothesis.
