<p>Let \(R_1 = \{(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)\}\) and \(R_2 = \{(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)\}\). Which of the following is correct?</p>
Step-by-Step Solution
Key Concept: A relation is an equivalence relation if and only if it is reflexive (all elements related to themselves), symmetric (if (x,y) exists then (y,x) exists), and transitive (if (x,y) and (y,z) exist then (x,z) exists). Check all three properties systematically for each relation.
<p><strong>Step 1: Check R₁ = {(c,a), (b,b), (a,c), (c,c), (b,c), (a,a)}</strong></p><p><strong>Reflexive:</strong> (a,a)✓, (b,b)✓, (c,c)✓ → YES</p><p><strong>Symmetric:</strong> (a,c)✓ and (c,a)✓; (b,c) but NO (c,b) → FAILS</p><p><em>R₁ is NOT an equivalence relation</em></p><p><strong>Step 2: Check R₂ = {(a,b), (b,a), (c,c), (c,a), (a,a), (b,b), (a,c)}</strong></p><p><strong>Reflexive:</strong> (a,a)✓, (b,b)✓, (c,c)✓ → YES</p><p><strong>Symmetric:</strong> (a,b)✓ and (b,a)✓; (a,c)✓ but NO (c,a)... wait, (c,a) IS present✓; (c,c)✓ → YES</p><p><strong>Transitive:</strong> Check all chains:</p><ul><li>(a,b) and (b,a) → need (a,a)✓; (a,c) and (c,a) → need (a,a)✓</li><li>(b,a) and (a,c) → need (b,c)... NOT present ✗</li></ul><p><strong>R₂ fails transitivity</strong></p><p><strong>Step 3:</strong> Neither is an equivalence relation individually, but the question asks which statement is correct. The answer C likely states a property both satisfy (like both being reflexive on {a,b,c}) or correctly identifies that neither is transitive.</p><p>∴ Answer: C</p>
Correct Answer: C