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

Dileep namdev
29/03/2023
0
Question-Answer

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.

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

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

Solution:-

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