<p>If \( A \) and \( B \) are two equivalence relations defined on set \( C \), then which of the following is always true?</p>
<p>(a) \( A \cap B \) is an equivalence relation</p>
<p>(b) \( A \cap B \) is not an equivalence relation</p>
<p>(c) \( A \cup B \) is an equivalence relation</p>
<p>(d) \( A \cup B \) is not an equivalence relation</p>
Step-by-Step Solution
Key Concept: The intersection of two equivalence relations is always an equivalence relation because it preserves reflexivity, symmetry, and transitivity. However, the union of two equivalence relations is NOT necessarily an equivalence relation as it may violate transitivity.
<p><strong>Step 1:</strong> Recall that an equivalence relation must satisfy reflexivity, symmetry, and transitivity.</p><p><strong>Step 2:</strong> Check <em>intersection</em> A ∩ B: If (x,y) ∈ A ∩ B, then (x,y) is in both A and B. Since both are reflexive, symmetric, and transitive individually, these properties are inherited by A ∩ B.</p><p><strong>Step 3:</strong> Verify each property for A ∩ B:</p><ul><li><strong>Reflexive:</strong> For any x ∈ C, (x,x) ∈ A and (x,x) ∈ B, so (x,x) ∈ A ∩ B ✓</li><li><strong>Symmetric:</strong> If (x,y) ∈ A ∩ B, then (x,y) ∈ A and (x,y) ∈ B. By symmetry in both, (y,x) ∈ A and (y,x) ∈ B, so (y,x) ∈ A ∩ B ✓</li><li><strong>Transitive:</strong> If (x,y), (y,z) ∈ A ∩ B, then both pairs are in A and in B. By transitivity in both, (x,z) ∈ A and (x,z) ∈ B, so (x,z) ∈ A ∩ B ✓</li></ul><p><strong>Step 4:</strong> The union A ∪ B fails: Take A = {(1,1), (2,2), (3,3), (1,2), (2,1)} and B = {(1,1), (2,2), (3,3), (2,3), (3,2)}. Then (1,2) ∈ A ∪ B and (2,3) ∈ A ∪ B, but (1,3) ∉ A ∪ B, violating transitivity.</p><p>∴ <strong>Answer: A ∩ B is always an equivalence relation</strong></p>
Correct Answer: A