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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

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    @RickDecker Sorry...my bad...question edited2012-06-07

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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 $z$

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    If you prove that it satisfies these three properties, then you prove it's an equivalence relation. Proving that these are true for equality will depend on whether you are taking equality as a defined relation or if it is a primitive.2012-06-07
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Now that we have the clarification from my comment below, your proof should be straightforward. Equality on any set is an equivalence relation.

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    I think he is saying $A=\{3,5,6,8,9,11,13\}$ and $R=\{(x,y)\in A^2|x=y\}$.2012-06-07