Check whether the relation R defined on R by R = (a, b): a b^3 is… - Vidyadip

Question

Check whether the relation $R$ defined on $R$ by $R =\left\{(a, b): a \leq b^3\right\}$ is reflexive, symmetric or transitive.

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Answer

Relation defined on $R$,
$S=\left\{(a, b): a \leq b^3\right\}$
For $a=\frac{1}{2},\left(\frac{1}{2}, \frac{1}{2}\right) \notin S \left(\because \frac{1}{2} \nless \frac{1}{8}\right)$
$\therefore \quad\left(\frac{1}{2}, \frac{1}{2}\right) \notin S \quad \therefore S$ is not reflexive.
Suppose, $(1,5) \in S$
Then, $(5,1) \notin S \quad(\because 5 \not \leq 1)$
$\therefore S$ is not symmetric.
Suppose, $(a, b) \in S$ and $(b, c) \in S$
$\begin{array}{l}
\therefore \quad a \leq b^3 \text { and } b \leq c^3 \\
\therefore \quad b^3 \leq c^9
\end{array}$
Thus, $a \leq b^3 \leq c^9$
$\begin{array}{ll}
\therefore & a \leq c^9 \\
\therefore & (a, c) \notin S
\end{array}$
$\therefore \quad S$ is not transitive.
Hence, S is not reflexive, symmetric, transitive.

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