R is a relation over set of real numbers and it is given by nm ge 0 . Then R is

English

Gujarati

Hindi

1.Relation and Function

easy

$R$ is a relation over the set of real numbers and it is given by $nm \ge 0$. Then $R$ is

Symmetric and transitive

Reflexive and symmetric

A partial order relation

An equivalence relation

Solution

Given, $m R n$ iff $m n \geq 0$

Reflexivity:

We know that $m^2 \geq 0$

$\Rightarrow mm \geq 0$

$\Rightarrow mRm$

Hence, $r$ is reflexive.

Symmetry:

Let $m R n$

$\Rightarrow m n \geq 0$

$\Rightarrow nm \geq 0$ (product of real numbers is commutative.)

$\Rightarrow nRm$

Hence, $R$ is symmetric.

Transitive

Let $m R n, n R p$

$\Rightarrow mn \geq 0 ; np \geq 0$

$\Rightarrow m , n , p$ are of same signs

$\Rightarrow mp \geq 0$

$\Rightarrow mRp$

Hence, $R$ is transitive.

Hence, $R$ is an equivalence relation.

Standard 12

Mathematics

Show that the relation $R$ defined in the set $A$ of all polygons as $R=\left\{\left(P_{1}, P_{2}\right):\right.$ $P _{1}$ and $P _{2}$ have same number of sides $\}$, is an equivalence relation. What is the set of all elements in $A$ related to the right angle triangle $T$ with sides $3,\,4$ and $5 ?$

medium

Let $R$ be a relation over the set $N × N$ and it is defined by $(a,\,b)R(c,\,d) \Rightarrow a + d = b + c.$ Then $R$ is

hard

Determine whether each of the following relations are reflexive, symmetric and transitive:

Relation $\mathrm{R}$ in the set $\mathrm{A}=\{1,2,3,4,5,6\}$ as $\mathrm{R} =\{(\mathrm{x}, \mathrm{y}): \mathrm{y}$ is divisible by $\mathrm{x}\}$

medium

The minimum number of elements that must be added to the relation $R =\{( a , b ),( b , c )\}$ on the set $\{a, b, c\}$ so that it becomes symmetric and transitive is:

hard

(JEE MAIN-2023)

Let $R_{1}$ and $R_{2}$ be two relations defined on $R$ by $a R _{1} b \Leftrightarrow a b \geq 0$ and $a R_{2} b \Leftrightarrow a \geq b$, then

medium

(JEE MAIN-2022)

$R$ is a relation over the set of real numbers and it is given by $nm \ge 0$. Then $R$ is

Solution

Similar Questions

Let $R$ be a relation over the set $N × N$ and it is defined by $(a,\,b)R(c,\,d) \Rightarrow a + d = b + c.$ Then $R$ is

Determine whether each of the following relations are reflexive, symmetric and transitive: Relation $\mathrm{R}$ in the set $\mathrm{A}=\{1,2,3,4,5,6\}$ as $\mathrm{R} =\{(\mathrm{x}, \mathrm{y}): \mathrm{y}$ is divisible by $\mathrm{x}\}$

The minimum number of elements that must be added to the relation $R =\{( a , b ),( b , c )\}$ on the set $\{a, b, c\}$ so that it becomes symmetric and transitive is:

Let $R_{1}$ and $R_{2}$ be two relations defined on $R$ by $a R _{1} b \Leftrightarrow a b \geq 0$ and $a R_{2} b \Leftrightarrow a \geq b$, then

Start a Free Trial Now

Determine whether each of the following relations are reflexive, symmetric and transitive:

Relation $\mathrm{R}$ in the set $\mathrm{A}=\{1,2,3,4,5,6\}$ as $\mathrm{R} =\{(\mathrm{x}, \mathrm{y}): \mathrm{y}$ is divisible by $\mathrm{x}\}$