Proof of equivalence relation

Question

I'm studying for my final exam of discrete mathematics, is an exercise in particular concerning equivalence relations do not know how to start:

$$ \text{Let } A = \left\{{3, 5, 6, 8, 9, 11, 13}\right\}\text{ and } R \subseteq A\times A: xRy\Longleftrightarrow{ x \equiv y}$$

How I can prove the symmetry, reflexivity and transitivity?

• $(1)$ symmetry ($xRx$ for any $x$),

• $(2)$ reflexivity ($xRy$ implies $yRx$), and

• $(3)$ transitivity ($xRy$ and $yRz$ implies $xRz$)

I know clearly that the properties must be satisfied by other exercises I've done, but this one in specific, I do not know how to prove mathematically

Answer 1

Now that we have the clarification from my comment below, your proof should be straightforward. Equality on any set is an equivalence relation.

Answer 2

This should follow from the fact that equality is an equivalence relation under any set.

$x=x$, for all $x$

$x=y \to y=x$, for all $x$ and $y$

if $x=y$ and $y=z$, then $x=z$, for any $x$, $y$, and $z$