A relation R on set A = \1, 2, 3, 4, 5\ is defined as R = \(x, y) : x² - y² < 8\. Check whether the relation R is reflexive, symmetric and transitive.

MathsCBSE Class 12 · Board exam 2024, set 65/1/1 · 3 marks · Short answer

CBSE Class 12 Mathematics 2024: question 26(a)

Set in the CBSE Class 12 board exam of 2024, paper 65/1/1, section C, question 26(a), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q26(a) · 3 marks · Short answerRelations and Functions

A relation R on set $A = {1, 2, 3, 4, 5}$ is defined as $R = {(x, y) : ∣ x^{2} - y^{2} ∣ < 8}$ . Check whether the relation R is reflexive, symmetric and transitive.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 65/1/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Check each property separately: (x,x), then (x,y)⇒(y,x), then the chain condition.

Hint 2

For symmetry use the fact that $∣ p ∣ = ∣ - p ∣$ .

Hint 3

For transitivity, look for a chain like (1,2),(2,3) and check whether $∣ 1^{2} - 3^{2} ∣ < 8$ .

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Reflexive: For every $x \in A$ , $∣ x^{2} - x^{2} ∣ = 0 < 8$ , so $(x, x) \in R$ . Hence R is reflexive.
½ mark
Symmetric: Let $(x, y) \in R$ . Then $∣ x^{2} - y^{2} ∣ < 8 \Rightarrow ∣ y^{2} - x^{2} ∣ = ∣ x^{2} - y^{2} ∣ < 8 \Rightarrow (y, x) \in R$ . Hence R is symmetric.
1 mark
Transitive: $(1, 2) \in R$ since $∣1 - 4∣ = 3 < 8$ , and $(2, 3) \in R$ since $∣4 - 9∣ = 5 < 8$ . But $∣1 - 9∣ = 8 \neq < 8$ , so $(1, 3) \in / R$ . Hence R is not transitive.
1½ marks

Answer: R is reflexive and symmetric but not transitive.

Where the marks are

Proves reflexive (0.5 mark)
Proves symmetric using $∣ x^{2} - y^{2} ∣ = ∣ y^{2} - x^{2} ∣$ (1 mark)
Counter-example (1,2),(2,3) ∈ R but (1,3) ∉ R, so not transitive (1.5 marks)

Revise the chapter

Chapter: Relations and Functions (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Relations and Functions, one at a time, marked against the answer key and marking points.Back to the full paper2024 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2024: question 26(a)

Hints

Step-by-step solution

Revise the chapter

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