Let A={2,3,6,8,9,11} and B={1,4,5,10,15} Let mathrm{R} be a relation on mathrm{A} × mathrm{B} defin

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1.Relation and Function

hard

Let $A=\{2,3,6,8,9,11\}$ and $B=\{1,4,5,10,15\}$

Let $\mathrm{R}$ be a relation on $\mathrm{A} \times \mathrm{B}$ define by $(\mathrm{a}, \mathrm{b}) \mathrm{R}(\mathrm{c}, \mathrm{d})$ if and only if $3 \mathrm{ad}-7 \mathrm{bc}$ is an even integer. Then the relation $\mathrm{R}$ is

reflexive but not symmetric.

transitive but not symmetric.

reflexive and symmetric but not transitive.

an equivalence relation.

(JEE MAIN-2024)

Solution

$ \mathrm{A}=\{2,3,6,8,9,11\} \quad(\mathrm{a}, \mathrm{b}) \mathrm{R}(\mathrm{c}, \mathrm{d}) $

$ \mathrm{B}=\{1,4,5,10,15\} \quad 3 \mathrm{ad}-7 \mathrm{bc} $

$ \text { Reflexive : }(\mathrm{a}, \mathrm{b}) \mathrm{R}(\mathrm{a}, \mathrm{b})$

$\Rightarrow 3 \mathrm{ab}-7 \mathrm{ba}=-4 \mathrm{ab}$ always even so it is reflexive.

Symmetric : If $3 \mathrm{ad}-7 \mathrm{bc}=$ Even

Case $-I$ : odd odd

Case $-II$ : even even

$(\mathrm{c}, \mathrm{d}) \mathrm{R}(\mathrm{a}, \mathrm{b}) \Rightarrow 3 \mathrm{bc}-3 \mathrm{ab}$

Case $-I$ : odd odd

Case $-II$ : even even

so symmetric relation

Transitive :

Set $(3,4) R(6,4)$ Satisfy relation

Set $(6,4) R(3,1)$ Satisfy relation

but $(3,4) R(3,1)$ does not satisfy relation so not transitive.

Standard 12

Mathematics

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

Relation $\mathrm{R}$ in the set $\mathrm{N}$ of natural numbers defined as

$\mathrm{R}=\{(x, y): y=x+5 $ and $ x<4\}$

medium

Let $R= \{(3, 3) (5, 5), (9, 9), (12, 12), (5, 12), (3, 9), (3, 12), (3, 5)\}$ be a relation on the set $A= \{3, 5, 9, 12\}.$ Then, $R$ is

hard

(JEE MAIN-2013)

Let a relation $R$ be defined by $R = \{(4, 5); (1, 4); (4, 6); (7, 6); (3, 7)\}$ then ${R^{ – 1}}oR$ is

medium

Check whether the relation $R$ defined in the set $\{1,2,3,4,5,6\}$ as $R =\{(a, b): b=a+1\}$ is reflexive, symmetric or transitive.

medium

Let $A=\{-4,-3,-2,0,1,3,4\}$ and $R =\{( a , b ) \in A$ $\times A : b =| a |$ or $\left.b ^2= a +1\right\}$ be a relation on $A$. Then the minimum number of elements, that must be added to the relation $R$ so that it becomes reflexive and symmetric, is $……..$.

medium

(JEE MAIN-2023)

Let $A=\{2,3,6,8,9,11\}$ and $B=\{1,4,5,10,15\}$ Let $\mathrm{R}$ be a relation on $\mathrm{A} \times \mathrm{B}$ define by $(\mathrm{a}, \mathrm{b}) \mathrm{R}(\mathrm{c}, \mathrm{d})$ if and only if $3 \mathrm{ad}-7 \mathrm{bc}$ is an even integer. Then the relation $\mathrm{R}$ is

Solution

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Let $A=\{2,3,6,8,9,11\}$ and $B=\{1,4,5,10,15\}$

Let $\mathrm{R}$ be a relation on $\mathrm{A} \times \mathrm{B}$ define by $(\mathrm{a}, \mathrm{b}) \mathrm{R}(\mathrm{c}, \mathrm{d})$ if and only if $3 \mathrm{ad}-7 \mathrm{bc}$ is an even integer. Then the relation $\mathrm{R}$ is

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

Relation $\mathrm{R}$ in the set $\mathrm{N}$ of natural numbers defined as

$\mathrm{R}=\{(x, y): y=x+5 $ and $ x<4\}$