Let $X$ a infinite set and $Y$ a finite set. Prove exists a function $f:X\rightarrow Y$ surjective and a function $g:Y\rightarrow X$ injective.

Question

Let $X$ a infinite set and $Y$ a finite set. Prove exists a function $f:X\rightarrow Y$ surjective and a function $g:Y\rightarrow X$ injective.

My work:

Let $f:X\rightarrow Y$ defined $f(x_i)=y_j$ such that $i\in [X]$ and $j\in[Y]$.

Go to prove $f$ is surjective.

Let $y_k\in Y$. As $f(x_i)=y_j$ with $x_i \in X$ and $y_i\in Y$ and $X$ is a infinite set. Then exists $x_k$ such that $f(x_k)=y_k$. In consequence $f$ is surjective.

Let $g:Y\rightarrow X$ defined $g(y_i)=x_j$ such that $i\in [Y]$ and $j\in[X]$. Let $y_1\,, y_2\in Y$ such that $g(y_1)=g(y_2)\implies x_1=x_2$. Here i'm stuck. Can someone help me?

Answer 1

You haven't actually defined a function at all - at least, not in any way that makes sense to a reader.

Hint: the way that I think best balances rigour and sensibleness-of-proof is to do it by induction on the size of the finite set.