# The Relation between Sets if I have a relation between A &#x00D7;<!-- × --> A if A

The Relation between Sets
if I have a relation between $A×A$
if $A=\left\{1,2,3\right\}$
if $B=\left\{1,2,3\right\}$
$R=\left\{\left(1,1\right),\left(2,2\right),\left(3,3\right)\right\}$
can I say that the Relation R is Reflexive and also a Symmetric because I have (a,b) and (b,a) and also (a,a).
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Step 1
The definition of the reflexive property is: For every .
We see that $A=\left\{1,2,3\right\}$, and for every variable it relates to itself given that the relation $R=\left\{\left(1,1\right),\left(2,2\right),\left(3,3\right)\right\}$.
Definition of antisymmetric is: whenever $a,b\in A$ are such that aRb and bRa , then necessarily $a=b$. With $A×A$, we see with our given R we see that this is antisymmetric.
Step 2
With the definition of symmetric: $\mathrm{\forall }a,b\in A\left(aRb\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}bRa\right)$, we see that this is symmetric.
So this relation is reflexive, antisymmetric and symmetric.