Relations & Functions
Types of Relations
Grade 12
Question:
<p>Let W denotes the words in the English dictionary. Define the relation R by <span>\(R = \{(x, y) \in W \times W : \text{the words } x \text{ and } y \text{ have atleast one letter in common}\}\)</span>. Then, R is</p>
<p>(a) reflexive, symmetric and not transitive</p>
<p>(b) reflexive, symmetric and transitive</p>
<p>(c) reflexive, not symmetric and transitive</p>
<p>(d) not reflexive, symmetric and transitive</p>
Step-by-Step Solution
Key Concept: Check each property separately: reflexivity (does each element relate to itself), symmetry (if xRy then yRx), and transitivity (if xRy and yRz then xRz).
<p><strong>Analysis:</strong> Let $W = \{\text{CAT}, \text{TOY}, \text{YOU}, \ldots\}$</p><p><strong>Reflexive:</strong> Any word has all its letters in common with itself, so $(x, x) \in R$ for all $x \in W$. Thus R is reflexive.</p><p><strong>Symmetric:</strong> If words $x$ and $y$ have at least one letter in common, then $y$ and $x$ also have that letter in common. Thus if $(x, y) \in R$ then $(y, x) \in R$. So R is symmetric.</p><p><strong>Transitive:</strong> Consider CAT and TOY: they share the letter T. TOY and YOU share the letters O and Y. But CAT and YOU share no common letters. So CAT R TOY, TOY R YOU, but CAT $\not$ R YOU. Thus R is not transitive.</p><p>∴ Answer is (a).</p>
Correct Answer: A