Sets, Relations & Functions
Types of Relations
Grade 11
Question:
<p>Given relation is <br>
\(R = \{(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)\}\)<br>
on set \(A = \{3, 6, 9, 12\}\). The relation \(R\) is:</p>
<p>(A) Reflexive and symmetric but not transitive</p>
<p>(B) Reflexive and transitive but not symmetric</p>
<p>(C) Symmetric and transitive but not reflexive</p>
<p>(D) An equivalence relation</p>
Step-by-Step Solution
Key Concept: A relation is transitive if whenever (a,b) and (b,c) are in R, then (a,c) must also be in R. Check this property systematically by examining each pair.
<p><strong>Step 1: Check Reflexivity</strong></p><p>For reflexivity, (a,a) must be in R for all a ∈ A. We have (3,3), (6,6), (9,9), (12,12) ∈ R. ✓ R is reflexive.</p><p><strong>Step 2: Check Symmetry</strong></p><p>For symmetry, if (a,b) ∈ R then (b,a) must be in R. We have (3,6) ∈ R but (6,3) ∉ R. ✗ R is not symmetric.</p><p><strong>Step 3: Check Transitivity</strong></p><p>Verify: if (a,b) and (b,c) ∈ R, then (a,c) ∈ R.</p><ul><li>(3,6) and (6,6) ∈ R ⟹ (3,6) ∈ R ✓</li><li>(3,6) and (6,12) ∈ R ⟹ (3,12) ∈ R ✓</li><li>(3,9) and (9,9) ∈ R ⟹ (3,9) ∈ R ✓</li><li>(3,12) and (12,12) ∈ R ⟹ (3,12) ∈ R ✓</li><li>All other pairs satisfy transitivity ✓</li></ul><p><strong>Conclusion:</strong> R is reflexive and transitive but not symmetric.</p><p>∴ Answer: B</p>
Correct Answer: B