CBSE Class 12 Mathematics 2023: question 32

Hints

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Step-by-step solution

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1. Reflexive: take $1\in \mathbb{R}$. Then $1\cdot 1=1$, which is rational, so $(1,1)\notin R$.

   Hence R is not reflexive.

   1½ marks

2. Symmetric: let $(x,y)\in R$, so $xy$ is irrational. Since $yx=xy$, $yx$ is also irrational, so $(y,x)\in R$.

   Hence R is symmetric.

   1½ marks

3. Transitive: $(1,\sqrt{2})\in R$ since $\sqrt{2}$ is irrational, and $(\sqrt{2},2)\in R$ since $2\sqrt{2}$ is irrational.

   But $1\cdot 2=2$ is rational, so $(1,2)\notin R$.

   Hence R is not transitive.

   2 marks

Answer: R is symmetric, but neither reflexive nor transitive.

Revise the chapter

• Chapter: Relations and Functions (notes, formulas, where marks are lost and every past question)
• Topic: Reflexive, Symmetric and Transitive Relations