why for all in definition of symmetric and transitive relation

Question

we know that a Relation $R$ on a Non empty set $A$ is said to be Reflexive if $(a,a) \in R$ $\forall a \in A$. For example if $A=\left\{1,2,3 \right\}$ then $R=\left\{(1,1),(2,2) \right\}$ is not Reflexive as $(3,3)$ is missing, since definition says $\forall a \in A$. But $R=\left\{(1,2),(2,1) \right\}$ is symmetric . But definition says $\forall a,b \in A$ if $(a,b) \in A$ then $(b,a) \in A$. But in above relation we have $a=1$,$b=2$,$a=2$,$b=1$. But we have not used $a,b$ as $3$. But still Relation is Symmetric. So why definition uses "For all" for symmetry and transitivity .

Answer 1

The definition of "symmetric" requires that for all $a,b$, if $(a,b)\in R$ then $(b,a)\in R$.

This is indeed true in the case where $a=b=3$: $$ \text{if }(3,3)\in R\text{ then }(3,3)\in R $$ or, for example, when $a=1$, $b=3$: $$ \text{if }(1,3)\in R\text{ then }(3,1)\in R $$ which is true because $(1,3)\in R$ is false, so there's no requirement for the "then" part to hold in that case.

See In classical logic, why is $(p\Rightarrow q)$ True if both $p$ and $q$ are False?.