Relation R ={( x , y ): x , y ∈ z and x + y is even } is :

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1.Relation and Function

medium

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

Areflexive and transitive but not symmetric

Breflexive and symmetric but not transitive

Can equivalence relation

Dsymmetric and transitive but not reflexive

(JEE MAIN-2025)

Solution

$R =\{( x , y ), x + y$ is even $x , y \in z \}$
reflexive $x+x=2 x$ even
symmetric of $x+y$ is even, then $(y+x)$ is also even
transitive of $x + y$ is even $\& y + z$ is even then $x+z$ is also even
So, relation is an equivalence relation.

Standard 12

Mathematics

How many reflexive relation are there on a set ' with $3$ elements

normal

Let $\mathrm{A}=\{1,2,3,4,5\}$. Let $\mathrm{R}$ be a relation on $\mathrm{A}$ defined by $x R y$ if and only if $4 x \leq 5 y$. Let $m$ be the number of elements in $\mathrm{R}$ and $\mathrm{n}$ be the minimum number of elements from $\mathrm{A} \times \mathrm{A}$ that are required to be added to $\mathrm{R}$ to make it a symmetric relation. Then $m+n$ is equal to:

hard

(JEE MAIN-2024)

Give an example of a relation. Which is Reflexive and transitive but not symmetric.

easy

Let $R$ be a relation on the set $N$ of natural numbers defined by $nRm $$\Leftrightarrow$ $n$ is a factor of $m$ (i.e.,$ n|m$). Then $R$ is