Question

If A = {a, b, c}, then the relation R = {(b, c)} on A is
(a)reflexive only
(b)symmetric only
(c)transitive
(d)reflexive and transitive only

Answer 1

Hint: For solving this problem, we consider all options individually. By using the necessary conditions for a set to be reflexive, symmetric and transitive, we proceed for solving the question. If any of the options fails to satisfy the condition, it would be rejected.

Complete step-by-step answer:
The conditions which must be true for a set to be reflexive, transitive and symmetric are:
1)For a relation to be reflexive, $\left( a,a \right)\in R$.
2)For a relation to be symmetric, $\left( a,b \right)\in R\Rightarrow \left( b,a \right)\in R$.
3)For a relation to be transitive, $\left( a,b \right)\in R,\left( b,c \right)\in R\Rightarrow \left( a,c \right)\in R$.
According to the problem statement, we are given a relation set R = {(b, c)} on A = {a, b, c}. For the set R, it should contain at least one element of the form (a, a) to be reflexive. Also, for being symmetric it should contain at least one element of the form (a, b) or (b, a). So, no element of these cases is possible, hence R is neither reflexive nor symmetry.
But R can be transitive because it contains only one element of the form (b, c) by using the above condition.
Therefore, option (c) is correct.

Note: The above relation R can also be proved transitive by using the below given formulation:
For all (x, y) in A, $\left( x,y \right)\in R,\left( y,z \right)\in R\Rightarrow \left( x,z \right)\in R$. Now let us try to prove this for set R on A.
 $\begin{align}
  & \left( x,y \right)=\left( y,z \right)=\left( b,c \right) \\
 & \therefore x=b,y=c=b,z=c,\text{ then }\left( x,z \right)=\left( b,c \right)\in R \\
\end{align}$
Thus, R is transitive.