Let R _{1}={( a , b ) ∈ N × N :| a - b | ≤ 13} and R _{2}={( a , b ) ∈ N × N :| a - b

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1.Relation and Function

hard

Let $R _{1}=\{( a , b ) \in N \times N 😐 a - b | \leq 13\}$ and $R _{2}=\{( a , b ) \in N \times N 😐 a - b | \neq 13\} .$ Thenon $N$

Both $R_{1}$ and $R_{2}$ are equivalence relations

Neither $R_{1}$ nor $R_{2}$ is an equivalence relation.

$R_{1}$ is an equivalence relation but $R_{2}$ is not

$R_{2}$ is an equivalence relation but $R_{1}$ is not

(JEE MAIN-2022)

Solution

$R_{1}=\{(a, b) \in N \times N:|a-b| \leq 13\}$

$R_{2}=\{(a, b) \in N \times N:|a-b| \neq 13\}$.

For $R_{1}$ :

$(i)\,Reflexive \,\,relation$

$(a, a) \in N \times N:|a-a| \leq 13$

$(ii)\, Symmetric\,\, relation$

$( a , b ) \in R _{1},( b , a ) \in R _{1}:| b – a | \leq 13$

$(iii) \,Transitive\, \,relation$

$( a , b ) \in R _{1},( b , c ) \in R _{1},( a , c ) \in R _{1}:$

$(1,3) \in R _{1,}(3,16) \in R _{1,} \text { but }(1,16) \notin R _{1}$

For $R _{2}$ :

$(i) \,Reflexive\,\, relation$

$(a, a) \in N \times N:|a-a| \neq 13$

$(ii)\, Symmetric\,\, relation$

$(b, a) \in N \times N:|b-a| \neq 13$

$(iii)\, Transitive \,\,relation$

$( a , b ) \in R _{2},( b , c ) \in R _{2},( a , c ) \in R _{2}$

$(1,3) \in R _{2,}(3,14) \in R _{2} \text { but }(1,14) \notin R _{2}$

Standard 12

Mathematics

Let $L$ be the set of all lines in $XY$ plane and $R$ be the relation in $L$ defined as $R =\{\left( L _{1}, L _{2}\right): L _{1} $ is parallel to $L _{2}\} .$ Show that $R$ is an equivalence relation. Find the set of all lines related to the line $y=2 x+4$

medium

Given a non empty set $X$, consider $P ( X )$ which is the set of all subsets of $X$.

Define the relation $R$ in $P(X)$ as follows :

For subsets $A,\, B$ in $P(X),$ $ARB$ if and only if $A \subset B .$ Is $R$ an equivalence relation on $P ( X ) ?$ Justify your answer.

hard

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medium

Let a relation $R$ on $\mathbb{N} \times \mathbb{N}$ be defined as : $\left(\mathrm{x}_1, \mathrm{y}_1\right) \mathrm{R}\left(\mathrm{x}_2, \mathrm{y}_2\right)$ if and only if $\mathrm{x}_1 \leq \mathrm{x}_2$ or $\mathrm{y}_1 \leq \mathrm{y}_2$

Consider the two statements :

($I$) $\mathrm{R}$ is reflexive but not symmetric.

($II$) $\mathrm{R}$ is transitive

Then which one of the following is true?

medium

(JEE MAIN-2024)

Let $R$ be a relation on the set $A$ of ordered pairs of positive integers defined by $(x, y) R (u, v)$ if and only if $x v=y u .$ Show that $R$ is an equivalence relation.

easy

Let $R _{1}=\{( a , b ) \in N \times N 😐 a - b | \leq 13\}$ and $R _{2}=\{( a , b ) \in N \times N 😐 a - b | \neq 13\} .$ Thenon $N$

Solution

Similar Questions

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Given a non empty set $X$, consider $P ( X )$ which is the set of all subsets of $X$.

Define the relation $R$ in $P(X)$ as follows :

For subsets $A,\, B$ in $P(X),$ $ARB$ if and only if $A \subset B .$ Is $R$ an equivalence relation on $P ( X ) ?$ Justify your answer.