What does "induced operations" means in congruence operations

Question

It says: prove that R is a congruence (it means it's a relation of equivalence and it preserves operations) ith respect to sum and multiplication in $\mathbb{R}$.

$a,b\in \mathbb{R}: aRb \iff a-b \in \mathbb{Z}$.

This is the problem of my question -> "Describe the induced operations in ${\mathbb{R}}/{R}$. Also what are their properties?"

I have a counterexample for multiplication, meaning this isn't a congruence respect to multiplication in $\mathbb{R}$, am I right?

What is the meaning of "induced operations" and how can I find them all, does square a number counts as operation?, how can I be sure i've found them all?.

Answer 1

For congruence or an equivalence relation, you need 3 properties:

• reflexive ($aRa$)
• symmetric ($aRb \Leftrightarrow bRa$)
• transitive ($aRb, bRc \Rightarrow aRc$)

In your case, $a-a=0 \forall a \in \mathbb{R}$, so $R$ is reflexive.

If $a,b \in \mathbb{R}$ and $a - b \in \mathbb{Z}$ then $b-a = -(a-b) \in \mathbb{Z}$, so $R$ is symmetric.

Can you prove $R$ is transitive?