Sets, Relations & Functions
Types of Relations
Grade 11
Question:
<p>Let <em>R</em> = {(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)} be a relation on the set <em>A</em> = {3, 6, 9, 12}. The relation is</p>
<p>reflexive and transitive only.</p>
<p>reflexive only.</p>
<p>an equivalence relation.</p>
<p>reflexive and symmetric only.</p>
Step-by-Step Solution
Key Concept: Check reflexivity (all elements map to themselves), transitivity (if aRb and bRc then aRc), and antisymmetry (if aRb and bRa then a=b) systematically. A relation satisfying all three is a partial order.
<p><strong>Step 1: Check Reflexivity</strong></p><p>For all a ∈ A, is (a,a) ∈ R?</p><p>✓ (3,3), (6,6), (9,9), (12,12) ∈ R. Reflexive ✓</p><p><strong>Step 2: Check Antisymmetry</strong></p><p>If (a,b) ∈ R and (b,a) ∈ R, is a = b?</p><p>Looking at R, no pair (a,b) with a≠b has both (a,b) and (b,a). For example, (3,6) ∈ R but (6,3) ∉ R. Antisymmetric ✓</p><p><strong>Step 3: Check Transitivity</strong></p><p>If (a,b) ∈ R and (b,c) ∈ R, is (a,c) ∈ R?</p><p>• (3,6) and (6,6) ⟹ (3,6) ✓</p><p>• (3,6) and (6,12) ⟹ (3,12) ✓</p><p>• (3,9) and (9,9) ⟹ (3,9) ✓</p><p>• (3,12) and (12,12) ⟹ (3,12) ✓</p><p>All transitive chains are satisfied. Transitive ✓</p><p><strong>Step 4: Conclusion</strong></p><p>R satisfies reflexivity, antisymmetry, and transitivity.</p><p>∴ <strong>Answer: A (Partial Order)</strong></p>
Correct Answer: A