Please elaborate on the concept of equivalence classes.

Question

I think I know hwo to prove reflexive, symmetric, and transitive properties but please show me how to do equivalence classes or describe them.

Let n ∈ N. Define relation Rn on Z by (x, y) ∈ Rn if and only if $x^2 − y^2$ is divisible by n. Prove that Rn is an equivalence relation. (a) Let n = 4. How many equivalence classes are there? Describe the equivalence classes in the simplest possible way. (b) Let n = 5. How many equivalence classes are there? Describe the equivalence classes in the simplest possible way.

Answer 1

Hint:

These relations say $x$ and $y$ are equivalent if and only if $x^2\equiv y^2\mod n$. Hence the equivalence classes corresp.ond to the congruence classes of the squares modulo $n$.

For instance, modulo $4$ the squares are congruent either to $0$ or to $1$. So there are two equivalence classes: even numbers have squares congruent to $0\bmod 4$ and odd numbers, squares congruent to $1 \bmod 4$.

Answer 2

Note that the number $n=5$ is a prime number. We want $x^2-y^2$ to be a multiple of $5$ which means $5$ should divide the product of the two numbers $x-y$ and $x+y$.

This boils down to one of them being a multiple of $5$, Choose any $x$. Then choose every possible $y$ such that $x-y$ or $x+y$ should be amultiple of $5$, By looking at the last digit of $x$, you can figure out how to list all such $y$'d for a given $x$.