Relations & Functions
Types of Relations
Grade 12
Question:
<p>Let \(R = \{(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)\}\) be a relation on the set \(A = \{3, 6, 9, 12\}\). The relation is</p>
<p>(a) reflexive and symmetric only</p>
<p>(b) an equivalence relation</p>
Step-by-Step Solution
Key Concept: An equivalence relation must be reflexive, symmetric, and transitive simultaneously. Failing any one property disqualifies it.
<p><strong>Analysis:</strong></p><p><strong>Reflexive:</strong> The pairs $(3,3), (6,6), (9,9), (12,12)$ are all in R, so every element is related to itself. R is reflexive.</p><p><strong>Symmetric:</strong> Check if whenever $(a,b) \in R$ we have $(b,a) \in R$. We have $(6,12) \in R$ but $(12,6) \notin R$. Therefore R is not symmetric.</p><p><strong>Transitive:</strong> Since R is not symmetric, we check transitivity. We have $(3,6) \in R$ and $(6,12) \in R$, but $(3,12) \in R$. Also $(3,9) \in R$ and $(9,9) \in R$, so $(3,9) \in R$ ✓. However, this alone doesn't establish full transitivity. Since $(6,12) \in R$ and we need $(12,y) \in R$ to check, but $(12,x) \notin R$ for $x \neq 12$, we cannot verify transitivity fully. But R fails symmetry, so it cannot be an equivalence relation.</p><p>∴ Answer is (a).</p>
Correct Answer: A