Mathematical notation for identity relation.

Question

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.

Answer 1

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*}