<p>Let <em>W</em> denote the words in the English dictionary. Define the relation <em>R</em> by: \(R = \{(x, y) \in W \times W\}\) the words \(x\) and \(y\) have at least one letter in common. Then <em>R</em> is</p>
<p>not reflexive, symmetric and transitive.</p>
<p>reflexive, symmetric and not transitive.</p>
<p>reflexive, symmetric and transitive.</p>
<p>reflexive, not symmetric and transitive.</p>
Step-by-Step Solution
Key Concept: Test reflexivity, symmetry, and transitivity by checking if the relation satisfies all three properties. Reflexivity fails because a word must relate to itself (trivially true), but transitivity fails since having one common letter with word x and one common letter with word y doesn't guarantee x and y share a common letter.
<p><strong>Step 1: Check Reflexivity</strong></p><p>For any word x ∈ W, does (x, x) ∈ R? Yes, word x shares all its letters with itself. ✓ Reflexive</p><p><strong>Step 2: Check Symmetry</strong></p><p>If (x, y) ∈ R, then x and y have at least one letter in common. This implies y and x have that letter in common, so (y, x) ∈ R. ✓ Symmetric</p><p><strong>Step 3: Check Transitivity</strong></p><p>If (x, y) ∈ R and (y, z) ∈ R, must (x, z) ∈ R?</p><p><strong>Counterexample:</strong> Let x = 'CAT', y = 'AXE', z = 'DOG'</p><ul><li>'CAT' and 'AXE' share letter 'A' → (CAT, AXE) ∈ R ✓</li><li>'AXE' and 'DOG' share letter 'E'... wait, they don't. Try: y = 'EGG'</li><li>'CAT' and 'EGG' share... they don't. Try: x = 'BAT', y = 'AXE', z = 'DOG'</li><li>'BAT' shares 'A' with 'AXE', 'AXE' shares 'E' with 'EGG', but 'BAT' and 'EGG' share no common letter</li></ul><p>✗ NOT Transitive</p><p><strong>Conclusion:</strong> R is reflexive and symmetric but NOT transitive, so it is <strong>NOT an equivalence relation</strong>.</p><p>∴ Answer: B (assuming B indicates: symmetric and reflexive, but not transitive)</p>
Correct Answer: B