I'm still a little mixed up on equivalence classes, so I'm trying to make some connections. I need to be specific of how many there are and what is in each.
Here's what I have:
Let $\mathscr R$ be the relation "congruence modulo $5$" on a set $A$, and let $a ∈ A$.
Then $[a] = \{ x\in \mathbb Z : 5\mid(x-a)\}$.
From my understanding, there are infinitely many equivalence classes:
$[0] = \{\dotsc, -5, 0, 5, 10, 15,\dotsc\}$
$[1] = \{\dotsc, -4, 1, 6, \dotsc\}$
…
$[5] = \{\dotsc, -5, 0, 5, 10, 15\dotsc\}$
…
Am I correct? How can I describe exactly what these classes contain using relations?
Best Answer
There would be 5 distinct equivalence classes for congruence modulo 5.These would be [0],[1],[2],[3],[4]. Notice how these classes together will cover all the integers. This is because the congruence class for
...[0]=[5]=[10]=...
...[1]=[6]=[11]=...
...[2]=[7]=[12]=...
...[3]=[8]=[13]=...
...[4]=[9]=[14]=...