Confusion about the definition of reflexive relation

Question

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?

Answer 1

Strictly speaking, the universal quantification should be there in all three cases. It's often just left out as a shorthand.

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Edit: strictly speaking, all statements are implicitly universally quantified. This is true even if the statement is written as an implication e.g.

$$\text{ if } a \in A \text{ then } (a,a) \in R$$

is really just

$$\forall a \in A, (a,a) \in R$$

or if you like, it is equivalent to the redundant statement:

$$\forall a \in A, \text{ if } a \in A \text{ then } (a,a) \in R$$

Otherwise, what is the $a$ that we are talking about? It needs to be drawn from some universe. So much of maths is implicitly universally quantified that we forget it's going on in the background. But from a strict, logical standpoint, it's always there.

Answer 2

Reflexive means EACH $a\in A$ is in relation with itself.

Symmetric means AS LONG AS $aRb$, then $bRa$.

Transitive means IF $aRb$, $bRc$, then $aRc$.

A relation that is reflexive, symmetric and transitive is called an equivalence relation. I would like to think "friendship" as an example of equivalent relation (in a perfect world in which 1) you are friend with yourself; 2) if you are friend to Tom, then Tom is a friend to you; 3) if you are friend to Tom, Tom is friend to Amy, then you are friend to Amy.)