R={(a, b): a ≤  b³} is reflexive, symmetric or transitive

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Question 2: Check whether the relation R in R defined as R={(a, b): a ≤  b³} is reflexive, symmetric or transitive.

Solution:-

R=\left\{(a, b): a \leq b^{3}\right\}

  Reflexive:- \left(\frac{1}{2}, \frac{1}{2}\right) \notin R, since \frac{1}{2}>\left(\frac{1}{2}\right)^{3}

\therefore \mathrm{R} is not reflexive.

Symmetric:- (1,2) \in R\left(\text { as } 1<2^{3}=8\right) \Rightarrow (2,1) \notin R\left(\text { as } 2^{3}>1=8\right)

R is not symmetric. 

Transitive:- \left(3, \frac{3}{2}\right),\left(\frac{3}{2}, \frac{6}{5}\right) \in R, \text { since } 3<\left(\frac{3}{2}\right)^{3} \text { and } \frac{2}{3}<\left(\frac{6}{2}\right)^{3}

\left(3, \frac{6}{5}\right) \notin R \Rightarrow 3>\left(\frac{6}{5}\right)^{3}

\therefore \mathrm{R} is not transitive.

\mathrm{R} is neither reflexive nor symmetric nor transitive.

Other question:-

R= {(a, b): a ≤ b²} is neither reflexive nor symmetric nor transitive

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