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

State whether the following graph is a one-to-one function or not, giving a reason.

State whether the following graph is a one-to-one function or not, giving a reason.

State whether the following graph is a one-to-one function or not, giving a reason.

Give an example of a relation which is reflexive, symmetric but not transitive.

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 $fog\left(\mathrm{x}\right)$.

Give an example of a relation which is reflexive, transitive but not symmetric

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.

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 $\left[2\right]$.