How do I interpret $\forall x \forall y R(x,y)$ given a set $M$

Question

Suppose I have a set $M = \{a,b\}$. Does $\forall x \ \forall y \ R(x,y)$ imply that $R = \{(a,b),(b,a),(a,a),(b,b)\}$ or does it imply that $R = \{(a,b),(b,a)\}$?

Answer 1

$\forall x \forall y R(x,y)$ means that $R$ relates every element in $M$ with every element (possibly the same) in $M$.

Thus $R=M\times M=\{(x,y)|x\in M \land y\in M\}$ (i.e. your 1st option for $R$).