Problem
Form relation inside set $ A = \{1,2,3\} $ so that $\text{R}:\text{A}\leftrightarrow \text{A}$
Attempt to solve
I know that $\text{R}:\text{A}\leftrightarrow \text{A}$ is true when relation is reflexive, symmetric and transitive. Which means when
$$ \underbrace{(\forall x \in A : xRx)}_{\text{relfexive}} \wedge \underbrace{(\forall x,y \in A : xRy \implies yRx )}_{\text{symmetric}}\wedge \underbrace{(\forall x,y,z \in A : (xRy \wedge yRz \implies xRz))}_{\text{transitive}}$$ $$ \implies \underbrace{\text{R}:\text{A}\leftrightarrow \text{A}}_{\text{equivalence relation}} $$
I could try to form binary relation explicitly $\forall x \in A$.
Example of symmetric binary relation:
$$ R = \{ \langle 1,1 \rangle, \langle 2,2 \rangle, \langle 3,3 \rangle \} $$
Example of symmetric relation, which is also transitive
$$ R = \{ \langle 1, 2 \rangle, \langle 2,3 \rangle \rangle, \langle 3 ,1 \rangle \}$$
So by combining these two i get relation that is equivalence relation?
$$ R = \{ \langle 1,1 \rangle, \langle 2,2 \rangle, \langle 3,3 \rangle ,\langle 1, 2 \rangle, \langle 2,3 \rangle \rangle, \langle 3 ,1 \rangle \} $$ I'am not quite sure if this is right? If someone could clarify on what is going on since i don't think i quite understand this yet.
For example I don't understand why reflexive + symmetric + transitive would make relation equivalence relation?
This (reflexive, symmetric, transitive) is just the definition of an equivalence relation.
Note that your relation is not an equivalence relation because if $(1,2)\in R$, then it must be true that $(2,1)\in R$ as well (similar for the other elements).
The easiest example of a non-empty equivalence relation over $A$ would be $\{(1,1)\}$; or the set containing all possible pairs in $A\times A$.