Embibe Experts Solutions for Chapter: Relations and Functions, Exercise 1: CBSE-2019

Let $\mathrm{A}=\{\mathrm{x}\in \mathrm{Z}:0\le \mathrm{x}\le 12\}$. Show that $\mathrm{R}=\left\{\right(\mathrm{a},\mathrm{b}):\mathrm{a},\mathrm{b}\in \mathrm{A},|\mathrm{a}-\mathrm{b}\left|\text{ is divisible by }4\right\}$ is an equivalence relation. Find the set of all elements related to $1$. Also write the equivalence class $\left[2\right]$.

Show that the function $f:R\to R$ defined by $f\left(x\right)=\frac{x}{x^2+1},\forall x\in \mathrm{ℝ}$ is neither one-one nor onto. Also, if $g:R\to R$ is defined as $g\left(x\right)=2 x-1$, find $fog\left(\mathrm{x}\right)$.

If $\mathrm{f}\left(x\right)=x+1$, find $\frac{\mathrm{d}}{\mathrm{d}x}\left(\text{ fof}\right)\left(x\right)$.

Show that the relation $R$ on $R$ is defined as $R=\left\{\right(a,b):a\le b\}$, is reflexive and transitive but not symmetric.

Prove that the function $f:N\to N$, defined by$f\left(x\right)=x^2+x+1$ is one-one but not onto. Find the inverse of $f:N\to S$, where $S$ is a range of $f$.

A relation $R$ in a set $A$ is called _____, if $\left(a_1,a_2\right)\in R$ implies $\left(a_2,a_1\right)\in R$, for all $a_1,a_2\in A$.

Consider a bijective function $f:R_{+}\to (7,\infty )$ given by $f\left(x\right)=16x^2+24x+7$, where $R_{+}$ the set of all positive real numbers. Find the inverse function of $f$.

Let $N$ be the set of natural numbers and $R$ be the relation on $N\times N$ defined by $(a,b)R(c,d) \mathrm{iff} ad=bc$ for all $a,b,c,d\in N$. Show that $R$ is an equivalence relation.