JEE MAIN - Mathematics (2006 - No. 30)
Let $W$ denote the words in the English dictionary. Define the relation $R$ by
$R=\{(x, y) \in W \times W \mid$ the words $x$ and $y$ have at least one letter in common}. Then, $R$ is
$R=\{(x, y) \in W \times W \mid$ the words $x$ and $y$ have at least one letter in common}. Then, $R$ is
reflexive, symmetric and not transitive
reflexive, symmetric and transitive
reflexive, not symmetric and transitive
not reflexive, symmetric and transitive
설명
Let's evaluate the relation $R$ for the properties of reflexivity, symmetry, and transitivity.
Relation R : $R={(x, y) \in W \times W \mid}$ the words $x$ and $y$ have at least one letter in common}.
- Reflexivity : Each word in English obviously has at least one letter in common with itself, so the relation is reflexive.
- Symmetry : If a word $x$ has at least one letter in common with a word $y$, then $y$ necessarily has that same letter in common with $x$. So, the relation is symmetric.
- Transitivity : This property is not always satisfied. For instance, consider the three words 'cat', 'bat', and 'bee'. 'Cat' and 'bat' share a letter (the 'a'), and 'bat' and 'bee' share a letter (the 'b'), but 'cat' and 'bee' do not share any letters. Therefore, even though 'cat' is related to 'bat' and 'bat' is related to 'bee', 'cat' is not related to 'bee', so the relation is not transitive.
In conclusion, the correct answer is Option A : $R$ is reflexive, symmetric and not transitive.