The relation R = {( x , y ) : x , y ∈ Z and x + y is even } is:

For reflexive ( x , x ) ∈ R , x ∈ Z ⇒ x + x = 2 x → even For symmetric of ( x , y ) ∈ R then ( y , x ) ∈ R when x , y ∈ Z x + y → even ⇒ y + x → even for transitive if ( x , y ) ∈ R ⇒ x + y → even ( y , z ) ∈ R ⇒ y + z → even x + 2 y + z → even ⇒ x + z is even ⇒ ( x , z ) ∈ R ⇒ R is an equivalence relation.

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MathematicsSets and RelationsQuestions on Symmetric Transitive and Reflexive PropertiesMedium2 minPYQ_2025

MathematicsMediumsingle choice

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

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Solution:

For reflexive $(x, x) \in R, x \in Z$ $\Rightarrow x + x = 2 x \to even$ For symmetric of $(x, y) \in R$ then $(y, x) \in R$ when $x, y \in Z$ $x + y \to$ even $\Rightarrow y + x \to even$ for transitive if $(x, y) \in R \Rightarrow x + y \to$ even $(y, z) \in R \Rightarrow y + z \to even$ $x + 2 y + z \to$ even $\Rightarrow x + z$ is even $\Rightarrow (x, z) \in R$ $\Rightarrow R$ is an equivalence relation.

Stream:JEESubject:MathematicsTopic:Sets and RelationsSubtopic:Questions on Symmetric Transitive and Reflexive Properties

⏱ 2mℹ️ Source: PYQ_2025

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