Class JEE Mathematics Sets, Relations, and Functions Q #1075
KNOWLEDGE BASED
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4 Marks 2006 AIEEE MCQ SINGLE
Let $W$ denote the words in the English dictionary. Define the relation $R$ by $R = {(x, y) ∈ W × W |$ the words $x$ and $y$ have at least one letter in common}. Then, $R$ is
(A) reflexive, symmetric and not transitive
(B) reflexive, symmetric and transitive
(C) reflexive, not symmetric and transitive
(D) not reflexive, symmetric and transitive
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Correct Answer: A
Explanation
To determine the properties of the relation $R$, we analyze reflexivity, symmetry, and transitivity.

Reflexivity: A word always shares at least one letter with itself. So, $(x, x) ∈ R$ for all $x ∈ W$. Therefore, $R$ is reflexive.

Symmetry: If a word $x$ has a letter in common with word $y$, then $y$ also has a letter in common with $x$. So, if $(x, y) ∈ R$, then $(y, x) ∈ R$. Therefore, $R$ is symmetric.

Transitivity: Consider the words 'cat', 'bat', and 'bee'. 'cat' and 'bat' share the letter 'a', so (cat, bat) ∈ R. 'bat' and 'bee' share the letter 'b', so (bat, bee) ∈ R. However, 'cat' and 'bee' do not share any common letters, so (cat, bee) ∉ R. Therefore, $R$ is not transitive.

Thus, the relation $R$ is reflexive, symmetric, and not transitive.

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