Not too sure how to approach this. Say I am given this:
All $x, y \in\mathbb{Z}, xRy \iff y = x + 6$ or $y = x - 6$
Question is to check whether this is an equivalence relation.
To approach this question, I presume I need to split into two cases due to $\iff$? For example:
1) $x, y \in\mathbb{Z}, xRy \Rightarrow y = x + 6$
2) $y = x - 6 \Rightarrow x, y \in\mathbb{Z}, xRy$
From here on, solve each of the above to make sure both get the same results - that is to prove reflexivity, symmetric and transitivity to conclude it is equivalence equation.
My confusion here is the use of $\iff$?. How does this affect how I find the 3 relations first? I haven't encounter $\iff$ before other than in logic.
The biconditional here (and often) means that it's the definition of the given relation $R$. In other symbols, considering relations as sets of ordered pairs, it would be $$R:=\{(x,y)\in\Bbb Z\times\Bbb Z\, :\, y=x+6\,\text{ or }\, y=x-6\}\,.$$ Note that this relation is symmetric but is neither reflexive nor transitive.