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

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1.Relation and Function

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

Relation $\mathrm{R}$ in the set $\mathrm{A}$ of human beings in a town at a particular time given by

$ \mathrm{R} =\{(\mathrm{x}, \mathrm{y}): \mathrm{x}$ and $ \mathrm{y}$ work at the same place $\}$

Option A

Option B

Option C

Option D

Solution

$\mathrm{R} =\{( \mathrm{x} , \mathrm{y} ): \mathrm{x}$ and $\mathrm{y}$ work at the same place $\}$

$\Rightarrow(\mathrm{x}, \mathrm{x}) \in \mathrm{R}$ [as $\mathrm{x}$ and $\mathrm{x}$ work at the same place $]$

$\therefore \mathrm{R}$ is reflexive.

If $(\mathrm{x}, \mathrm{y}) \in \mathrm{R},$ then $\mathrm{x}$ and $\mathrm{y}$ work at the same place.

$\Rightarrow \mathrm{y}$ and $\mathrm{x}$ work at the same place.

$\Rightarrow(\mathrm{y}, \mathrm{x}) \in \mathrm{R}$

$\therefore \mathrm{R}$ is symmetric.

Now, let $(\mathrm{x}, \mathrm{y}),\,(\mathrm{y}, \mathrm{z}) \in \mathrm{R}$

$\Rightarrow \mathrm{x}$ and $\mathrm{y}$ work at the same place and $\mathrm{y}$ and $\mathrm{z}$ work at the same place.

$\Rightarrow \mathrm{x}$ and $\mathrm{z}$ work at the same place.

$\Rightarrow(\mathrm{x}, \mathrm{z}) \in \mathrm{R}$

$\therefore \mathrm{R}$ is transitive.

Hence. $\mathrm{R}$ is reflexive, symmetric, and transitive.

Standard 12

Mathematics

Let $N$ denote the set of all natural numbers and $R$ be the relation on $N \times N$ defined by $(a, b)$ $R$ $(c, d)$ if $ad(b + c) = bc(a + d),$ then $R$ is

hard

Let $R_1$ and $R_2$ be two relations on a set $A$ , then choose incorrect statement

normal

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)

The relation $R =\{( x , y ): x , y \in z$ and $x + y$ is even $\}$ is :

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(JEE MAIN-2025)

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medium

(JEE MAIN-2023)

Determine whether each of the following relations are reflexive, symmetric and transitive: Relation $\mathrm{R}$ in the set $\mathrm{A}$ of human beings in a town at a particular time given by $ \mathrm{R} =\{(\mathrm{x}, \mathrm{y}): \mathrm{x}$ and $ \mathrm{y}$ work at the same place $\}$

Solution

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

Relation $\mathrm{R}$ in the set $\mathrm{A}$ of human beings in a town at a particular time given by

$ \mathrm{R} =\{(\mathrm{x}, \mathrm{y}): \mathrm{x}$ and $ \mathrm{y}$ work at the same place $\}$