Sets, Relations & Functions
Types of Relations
Grade 11
Question:
<p>Let \(W\) be the set of words in the English dictionary. Define the relation \(R = \{(x, y) \in W \times W\}\) such that \(x\) and \(y\) 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: A relation is reflexive if every element relates to itself, symmetric if x~y implies y~x, and transitive if x~y and y~z implies x~z. Here, every word shares letters with itself (reflexive), sharing is mutual (symmetric), but transitivity fails: 'cat' and 'art' share 'a', 'art' and 'ten' share 't', but 'cat' and 'ten' share no letters.
<p><strong>Step 1: Check Reflexivity</strong> Does every word relate to itself? Yes—any word x shares all its letters with itself. ✓ Reflexive</p><p><strong>Step 2: Check Symmetry</strong> If word x shares a letter with word y, does y share a letter with x? Yes, by definition of 'common letter' the relation is mutual. ✓ Symmetric</p><p><strong>Step 3: Check Transitivity</strong> If x shares a letter with y AND y shares a letter with z, must x share a letter with z? <strong>No.</strong> Counterexample: x='cat', y='art', z='ten'. Here 'cat' and 'art' share 'a', 'art' and 'ten' share 't', but 'cat' and 'ten' share no common letter. ✗ Not Transitive</p><p><strong>Conclusion:</strong> R is reflexive and symmetric but not transitive. R is <strong>reflexive and symmetric only.</strong></p><p>∴ Answer: B</p>
Correct Answer: B