Check whether the relation \(\mathrm{R}\) in \(\mathbf{R}\) defined by \(\mathrm{R}=\left\{(a, b): a \leq b^3\right\}\) is reflexive, symmetric or transitive.
Expert Answer
Solution: \(R=\left\{(a, b): a \leq b^3\right\}, A=(-\infty, \infty)\) or \(R\)
(a) Reflexive : \(R=\left\{(a, a): a \leq a^3 \Rightarrow a\left(a^2-1\right) \geq 0\right.\)
This is not true of all \(a \in A\). So not reflexive.
(b) Symmetric : \(R=\left\{\left(a_1, a_2\right): a_1 \leq a_2^3\right\}\) \(R=\left\{\left(a_2, a_1\right): a_2 \leq a_1^3\right\}\) so not symmetric.
(c) Transitive : \(R=\left\{\left(a_1, a_2\right): a_1 \leq a_2^3\right\}\)
\[
R=\left\{\left(a_2, a_3\right): a_2 \leq a_3^3\right\} \Rightarrow a_2^3 \leq a_3^9
\]
So, \(a_1 \leq a_3^9 \nRightarrow a_1 \leq a_3^3\). Not transitive.
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