Give an example of a relation. Which is Reflexive and transitive but not symmetric.
Give an example of a relation. Which is Reflexive and transitive but not symmetric.
Define a relation $R$ in $R$ as $:$
$\left.R=\{a, b): a^{3} \geq b^{3}\right\}$
Clearly $(a,a)\in R$ as $a^{3}=a^{3}$
$\therefore R$ is reflexive.
Now, $(2,1)\in R$ $[$ as $2^{3} \geq 1^{3}]$
But, $(1,2)\notin R$ $[$ as $1^{3} < 2^{3}]$
$\therefore R$ is not symmetric.
Now, Let $(a, b),\,(b, c) \in R$
$\Rightarrow a^{3} \geq b^{3}$ and $b^{3} \geq c^{3}$
$\Rightarrow a^{3} \geq c^{3}$
$\Rightarrow(a, c) \in R$
$\therefore R$ is transitive.
Hence, relation $R$ is reflexive and transitive but not symmetric.
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Let $R_{1}$ and $R_{2}$ be relations on the set $\{1,2, \ldots, 50\}$ such that $R _{1}=\left\{\left( p , p ^{ n }\right)\right.$ : $p$ is a prime and $n \geq 0$ is an integer $\}$ and $R _{2}=\left\{\left( p , p ^{ n }\right)\right.$ : $p$ is a prime and $n =0$ or $1\}$. Then, the number of elements in $R _{1}- R _{2}$ is........
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