Mathematical notation for identity relation. For an identity relation, are the following mathematical notations correct?
∀a,b ∈ A : aRa ∧ ∄ aRb : a ≠ b
R = { (a,a) : a ∈ A }
What i want to say in first notation is for every a and b in A, a is related to itself, and there does not exist a relation between a and b, where a is not equal to b.
 A: The second one is definitely correct.  You can define a relation by its pairs.
$$
    R = \left\{(a,a) : a \in A \right\}
$$
Reading the first one, it looks like you're saying that $a$ is related to $a$ for every $a$, and there is no $b$ not equal to $a$ that is related to $a$.  That might look more like this:
$$
    \forall a \in A,\left( a \mathrel{R} a\right)
    \wedge \neg \left(\exists b \in A,\ (a \neq  b) \wedge (a \mathrel{R} b)\right)
$$
That second part can be simplified logically, though.
\begin{align*}
    \neg \left(\exists b \in A,\ (a \neq  b) \wedge (a \mathrel{R} b)\right)
    &\equiv \forall b \in A, (a=b) \vee (a \not\mathrel{R} b) \\
    &\equiv \forall b \in A, (a \mathrel{R} b) \implies (a = b)
\end{align*}
That last part is because $p \implies q$ is equivalent to $(\neg p) \vee q$.
The first part can be combined with it:
\begin{align*}
    \forall a \in A,\ a \mathrel{R} a 
    \equiv \forall a \in A,\ \forall b \in A,\ \left(a = b \implies a \mathrel{R} b\right)
\end{align*}
So the two together say:
\begin{align*}
    &\forall a \in A,\ \forall b \in A,\ 
    \left((a = b) \implies (a \mathrel{R} b)\right)
    \wedge
    \left((a \mathrel{R} b) \implies (a = b)\right)
    \\&\equiv\forall a \in A,\ \forall b \in A,\ 
    \left((a = b) \iff (a \mathrel{R} b)\right)
\end{align*}
