Injective function from N/R to {x,y}

Question

Given the equivalence relation on N aRb $\leftrightarrow $ a and b are both even or odd, given a function $f: N\rightarrow\{x,y\}$ with $f(n)=x$ if n is even and $f(n)=y$ if n is odd exists an injective function from $N/R$ to $\{x,y\}$? Can I use a homeomorphism theorem?

Answer 1

You need to show that $\Bbb N/R$ has only two elements. Then you can just create an injective function by taking one to $x$ and the other to $y$.

Answer 2

I have two different classes of equivalence : {[0],[1]}.

So can I defined an injective function F so that F{[a]}= x if [a]=[0] and y if [a]=[1]?