How to prove $R=\{(x,y)\mid x ,y\} $ is an equivalence relation?

Question

$R=\{(x,y)\mid x,y\; \text{are 5 bit strings with equal numbers of zeros }\}$ Prove that $R$ is an equivalence relation. How much classes does it have?

Answer 1

Consider the function $f$ that assigns the number of zeroes to its input which is a bit string of length $5$.

It's straightforward that $R$ is just the 'kernel' of $f$, i.e. $(x,y)\in R\iff f(x)=f(y) $.

The number of equivalence classes of $R$ are thus corresponding to the possible values (=elements of the range) of $f$, which in this case is $\{0,1,2,3,4,5\}$.