Question

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

Hint: We will be using the concepts of functions and relations to solve the problem. We will be using the definitions of reflexive relation, symmetric relations and transitive relations to verify if each relation holds or not and hence deduce the answer.

Complete step-by-step answer:

Now, we have been given a relation and we have to find whether the relation is reflexive, symmetric, transitive or a combination of these.
Now, we know that reflexive relations are those in which every element is mapped to itself i.e. $\left( a,a \right)\in R$ while symmetric relations are those for which if a R b then b R a. Also, holds and transitive are those relations in which if a R b and b R c then a R c must be held.
Now, we know different types of relations, we will check the given relation for these.
Now, we have been given a relation R by$R=\left\{ \left( x,y \right)\in W\times W\ the\ word\ x\ and\ y\ have\ at\ least\ one\ letter\ in\ common \right\}$
Now, for the relation to be reflexive. We have a word x, now a word x will have every letter common. Therefore, $\left( x,x \right)\in R$ and R is a reflexive relation.
Now, for the relation to be symmetric we have,
$\left( x,y \right)\in R$ i.e. x and y have at least one letter common.
We can write this as y and x have at least one letter common. So,
$\left( y,x \right)\in R$
$\Rightarrow \left( x,y \right)\in R\Rightarrow \left( y,x \right)\in R$ and hence, R is symmetric.
Now, for the relation to be transitive. We have if,
$\left( x,y \right)\in R$ i.e. x and y have at least one letter common.
$\left( y,z \right)\in R$ i.e. y and z have at least one letter common.
Now, it not necessary that x and z have a letter in common for example, if we take x = AND, y = NEXT and z = HER, then $\left( x,y \right)\in R$ and $\left( y,z \right)\in R$ but $\left( x,z \right)\notin R$.
Hence, the given relation is reflexive, symmetric and not transitive.
So, the correct answer is (B).

Note: To solve these types of questions it is important to note that a R b means that a is related to b by a relation R. Also these types of questions are solved easily by giving examples and counterexamples. Also, we have to check the relation for reflexive, symmetric and transitive relation to check it for equivalence relation.