What does it mean to calculate a relation's quotient set (the set of all of equivalence classes)?

Question

The set in question: T = {(a,a),(b,b),(c,c)}.

I am confused what it means by this, and I haven't found any resources online that helps explain this to me well enough. Any help is much appreciated.

Answer 1

Given a set such as $S=${$a,b,c$} and an equivalence relation on it, such as $T$,

the equivalence class of an element, say $a$, is the set {$x\in S|(a,x)\in T$} of what is related to $a$.

For the relation $T=${$(a,a),(b,b),(c,c)$}, the equivalence class of an element, say $a$, is simply {$a$}.

($b$ and $c$ are not related to $a$.)

The quotient set is the set of all equivalence classes.

The quotient set of $T=${$(a,a),(b,b),(c,c)$} is simply {{a}, {b}, {c}}.