Question

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

• A. reflexive, not symmetric and transitive
• B. not reflexive, symmetric and transitive
• C. reflexive, symmetric and not transitive
• D. reflexive, symmetric and transitive

Answer 1

Answer: (C)

$(x, x) \in R \quad \forall x \in W$ as all letters in both are common. If $(x, y) \in R$ then $x$ and $y$ have a letter in com$\operatorname{mon} \Rightarrow(y, x) \in R$.
Next, let $x=$ fix, $y=\operatorname{six}$ and $z=\operatorname{son}$ then $(x, y) \in R$, $(y, z) \in R$ but $(x, z) \notin R$
So $R$ is reflexive, symmetric but not transitive