Difference between $a,b \in S$ and $\forall a,b \in S$

Question

Is there any difference between these two notations,

$a,b \in S$

$\forall a,b \in S$ where $S$ is any non-empty set.

Answer 1

Strictly speaking the symbol $a,b\in S$ on its own means only that $a$ and $b$ are in $S$. $a$ and be $b$ need to have been specified earlier in order to determine where they live to begin with.

For instance, in the sentence "$ab\in S$, for $a,b\in S$", clearly, the symbol "for $a,b\in S$" stands for $\forall a,b\in S$. Some people may adopt the notation $a,b\in S$ in place of $\forall a,b\in S$, for brevity. You need to talk to your teacher to see if that is the case.