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Let $W$ denote the words in the English dictionary. Define the relation $R$ by:
$\mathbf{R}=\{(x, y) \in W \times W$ the words $x$ and $y$ have at least one letter in common\}. Then $R$ is

1) not reflexive, symmetric and transitive
2) reflexive, symmetric and not transitive
3) reflexive, symmetric and transitive
4) reflexive, not symmeric and transitive
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## 1 Answer

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Answer

(2)

Explanation

Clearly $(x, x) \in R \forall x \in W .$ So, $\mathrm{R}$ is reflexive
Let $(x, y) \in R$, then $(y, x) \in R$ as $x$ and $y$ have
at least one letter in common. So, $R$ is symmetric.
But $R$ is not transitive for example Let $x=$ DELHI, $y=$ DWARKA and $z=$ PARK
then $(x, y) \in R$ and $(y, z) \in R$ but $(x, z) \notin R$.

by Diamond (74,866 points)

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