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Reflexive Relations

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Question

Check whether the relation $R$ defined on the set of real numbers, $R = {(a, b) : a \leq b^{3}}$ is reflexive, symmetric or transitive.

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Solution

Given the relation $R$ defined on the set $R = {(a, b) : a \leq b^{3}}$ .
This relation is not reflexive as $(- 2, - 2) \notin R$ as $- 2 > - 8$ .
Again this relation is not symmetric as $(1, 2) \in R$ but $(2, 1) \notin R$ since $2 > 1$ .
Similarly the relation $R$ is not transitive as $(26, 3) \in R, (3, 2) \in R$ but $(26, 2) \notin R$ .

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