The definition of a reflexive relation over $A$ is:
$R$ is reflexive over $A$ iff $\forall a \in A :(a,a) \in R$
Why the '$\forall a \in A$'? Def. of transitive and symmetric relations don't have that:
$R$ is transitive iff $(a,b)\in R \wedge (b,c) \in R \implies (a,c) \in R$
$R$ is symmetric iff $(a,b) \in R \implies (b,a) \in R$
If the universal quantification is needed to define the reflexive relation, why isn't it needed for the other two?