Relations & Functions
Types of Relations
Grade 12
Question:
<p>Let W denote the words in the English dictionary. Define the relation R by \(R = \{(x, y) \in W \times W\}\) the words <i>x</i> and <i>y</i> have at least one letter in common, then R is</p>
<p>(a) not reflexive, symmetric and transitive</p>
<p>(b) reflexive, symmetric and not transitive</p>
<p>(c) reflexive, symmetric and transitive</p>
<p>(d) reflexive, not symmetric and transitive</p>
Step-by-Step Solution
Key Concept: Check relation properties with concrete word examples; transitivity fails with the letter-sharing relation
<p><strong>Analysis:</strong> Reflexivity: Every word has at least one letter in common with itself ✓. Symmetry: If <i>x</i> and <i>y</i> share a letter, then <i>y</i> and <i>x</i> share the same letter ✓. Transitivity: Consider "car" and "arc" (share letters), "arc" and "cat" (share letters), but "car" and "cat" may not share all necessary letters consistently. Counterexample: "car" shares letters with "rat", "rat" shares letters with "tea", but "car" and "tea" do not share letters ✗.</p><p>∴ Answer is (b).</p>
Correct Answer: b