I want to calculate the residue class in $\mathbb{Z}/7\mathbb{Z}$, when $a = \{ b ~|~ b \equiv_7 a\}$.
Is my following calculation right?
$\bar{2} \cdot \bar{3} + \bar{4} \cdot \bar{3} = \bar{6} + \bar{12} = \bar{18} - \bar{14} = \bar{4}$
This feels like a simple calculation but I have trouble understanding this part " when $a = \{ b ~|~ b \equiv_7a\}$. "
Ty in advance.
I guess this definition wants to introduce the bar notation, i.e. $$\bar a\,:=\,\{b\in\Bbb Z:b\equiv_7 a\}\,.$$ And the important thing is that the operations $+,-,\cdot$ are well defined on these elements, meaning that $a\equiv_7a',\ b\equiv_7b'\implies a\cdot b\equiv_7a'\cdot b'$, and similarly for the other operations.
So it doesn't matter which representatives you execute the operation on, you will always end up in the same residue class.
Your calculations seem correct.