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:- 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:-

Question 1:-  Show that the relation R on the set R of real numbers, defined as R= {(a, b): a ≤ b²} is neither reflexive nor symmetric nor transitive.

Solution: See full solution

Question 3:-    Consider \large f:R_+ \rightarrow [-9, \infty) given by \large f(x) = 5x^2+6x-9 prove that f is bijective.

Solution: See full solution

Question 4:- Consider f:R_+ → [4, ∞) given by f(x) = x² + 4. Show that f is invertible.         [CBSE(AI) 2013]

Solution: See full solution

Question 5:-  Let f : R – {-4/3} → R be a function defined as f(x) = 4x/(3x+4). Show that, f:R – {-4/3} → Range of f, f is one-one and onto.                      [CBSE  2017(C)]

Solution:- See full solution

Question 6:- Let A = R – {3}, B = R – {1}. If f : A → B be defined by f(x) = \dfrac{x - 2}{x - 3}, ∀x ∈ A. Then, show that f is bijective.

Solution:- See full solution

Question 7:- Let f : W → W, be defined as f(x) = x – 1, if x is odd and f(x) = x + 1, Show that f is bijective.

Solution: See full solution

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