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?
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$(1)$ symmetry ($xRx$ for any $x$),
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$(2)$ reflexivity ($xRy$ implies $yRx$), and
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$(3)$ 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
Best Answer
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 $z$