For $m,n$ in $N$ define $m$ equivalent $n$ if $m^2-n^2$ is multiple of $3$
a) show that this is an equivalence relation
b) list elements in equivalence class [0]
c) list elements in equivalence class [1]
d)do you think there are any more equivalence classes?
for part a i proved that its true by showing through matrix that its symmetric,reflexive and transitive. my problem is that i don't really understand how to find equivalence class, can someone explain how to do it and show on part b or c, rest ill do myself once i get how to do it.
Best Answer
There are exactly two equivalence classes:
a. The set of integers divisible by 3, and
b. The set of integers non-divisible by 3.
Clearly, if $3\mid m$ and $3\mid n$, then $3\mid m^2$ and $3\mid n^2$, and hence $3\mid m^2-n^2$.
If $3\not\mid m$ and $3\not\mid n$, then $m^2$ and $n^2$ leave remainder 1, when divided by 3, and hence $3\mid m^2-n^2$.