R D Sharma Solutions for Chapter: Relations, Exercise 3: EXERCISE 2.3

Let $A$ be the set of first five natural numbers and let $R$ be a relation on $A$ defined as follows:

$\left(x,y\right)\in R⟺x\le y$

Express $R$ and $R^{-1}$ as sets of ordered pairs. Determine also the range of $R$.

Let $A=\left\{1\right\}$ and $B=\{3,4,5\}$. Find the total number of relations from $A$ to $B$.

Let $A=\{x,y,z\}$ and $B=\left\{a,b\right\}$. Find the total number of relations from $A$ to $B$.

Let $A=\left\{1,2,3,\mathrm{4,5},6\right\}$. Let $R$ be a relation on $A$ defined by

$R=\left\{\left(a,b\right): a,b\in A, b \mathrm{i}\mathrm{s} \mathrm{e}\mathrm{x}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{l}\mathrm{y} \mathrm{d}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{b}\mathrm{l}\mathrm{e} \mathrm{b}\mathrm{y} a\right\}$

Find the domain of $R$

Let $A=\left\{1,2,3,\mathrm{4,5},6\right\}$. Let $R$ be a relation on $A$ defined by

$R=\left\{\left(a,b\right): a,b\in A, b \mathrm{i}\mathrm{s} \mathrm{e}\mathrm{x}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{l}\mathrm{y} \mathrm{d}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{b}\mathrm{l}\mathrm{e} \mathrm{b}\mathrm{y} a\right\}$

Find the range of $R$

Figure shows a relationship between the sets $P$ and $Q$. Write this relation in

Roster form. What is its domain and range?

Let $R$ be a relation on $N\times N$ defined by

$\left(a,b\right)R\left(c,d\right)\iff a+d=b+c$ for all $\left(a,b\right),\left(c,d\right)\in N\times N$

Show that:

$\left(a,b\right)R\left(c,d\right)\Rightarrow \left(c,d\right)R\left(a,b\right)$ for all $\left(a,b\right),\left(c,d\right)\in N\times N$

Let $R$ be a relation on $N\times N$ defined by

$\left(a,b\right)R\left(c,d\right)\iff a+d=b+c$ for all $\left(a,b\right),\left(c,d\right)\in N\times N$

Show that:

$(a,b)R(c,d)$ and $\left(c,d\right)R\left(e,f\right)\Rightarrow (a,b)R(e,f)$ for all $\left(a,b\right),\left(c,d\right),\left(e,f\right)\in N\times N$