Equivalence relation and class

Question

Let X = {1, 2, 4, 7, 8, 10, 12, 14, 15, 18, 22, 23}. Consider the relation R on X given by (a, b) ∈ R if a − b is divisible by 7.

(a) Show that R is an equivalence relation on X.

(b) Determine the equivalence class [1].

(c) Determine the partition of X into equivalence classes.

edit 1) I need help on C, I get a) is congruent modulo and therefore b) is "the quotient of X modulo R" by my understanding but I don't quite understand how to write out the notation of the partition in this regard. Thank you for the quick answers, sorry I was not immediately specific.

edit 2) Thank for the explanations and help in the comments, especially JMoravitz and Lee Mosher. Much appreciated!

Answer 1

To answer C, you can do things one step at a time.

Starting with $1 \in X$, to determine the equivalence class $[1]$, ask yourself this question:

• What elements $a \in X$ are equivalent to $1$?

Then translate:

• For what elements $a \in X$ is it true that $(a,1) \in R$?

Then translate again:

• For what elements $a \in X$ is it true that $a-1$ is divisible by $7$?

Now go through the elements $a \in X$ one-by-one, answering this question for each in turn:

• For $a=1$, is $a-1=1-1=0$ divisible by $7$? Yes it is, so $1 \in [a]$.
• For $a=2$, is $a-1=2-1=1$ divisible by $7$? No it is not, so $2 \not\in [a]$.
• For $a=4$, is $a-1=4-1=3$ divisible by $7$? No it is not, so $4 \not\in [a]$.

And so on.

Along the way, you may have a leap of understanding which let's you cut off all of this one-by-one stuff. But until you do, keep doing the one-by-one stuff.