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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?

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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 $A\times A$.

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