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 in common}

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

Answer 1

The correct option is B

Reflexive, symmetric, and not transitive

Explanation for the correct answer:

The given relation is $R=\{(x,y)\in W\times W|$the words $x$ and $y$ have at least one letter in common}

Checking for Reflexive Property:

$(a,a)\in R$

If $a\in W,a=RAM$ (say)

$R$ is reflexive

Checking for Symmetric Property:

$$\begin{gathered} (RAM,AAM)\in R \\ (AAM,RAM)\in R \end{gathered}$$

If $(a1,a2)\in R$ and $R$$(a2,a1)\in R$, then R is symmetric.

Checking for Transitive Property

If$(a1,a2)\in R$and $(a2,a3)\in R$, then $(a1,a3)\in R$

$$\begin{gathered} a1=RAM,(a1,a2)\in R \\ a2=NAAM,(a2,a3)\in R \\ a3=NONE,(a1,a3)\notin R \end{gathered}$$

R is not transitive.

Hence, the correct option is (B)