Question

If A = \left\\{ {x,y,z} \right\\} then the relation R = \left\\{ {\left( {x,x} \right),\left( {y,y} \right),\left( {z,x} \right),\left( {z,y} \right)} \right\\} is
(A) Symmetric
(B) Transitive
(C) None of these
(D) Both A and B

Answer 1

In order to find the given relation is reflexive, symmetric, or transitive, we first need to understand the definition of the reflexivity, symmetry and transitivity holds for a set. The relation is reflexive if $\left( {x,x} \right)$ belongs to the relation for all $x$ belongs to the set. The relation is symmetric if $\left( {x,y} \right)$ belongs to the relation implies that $\left( {y,z} \right)$ belongs to the same relation. And the relation is transitive if $\left( {x,y} \right)\& \left( {y,z} \right)$ belongs to the relation implies that $\left( {x,z} \right)$ belongs to the same relation.

Complete step by step solution:
The relation R in A is said to be reflexive, if $\left( {a,a} \right) \in R {\text{ for }} a \in A$.
The relation R in A is said to be symmetric, if $\left( {a,b} \right) \in R \Rightarrow \left( {b,a} \right) \in R {\text{ for }} a, b \in A$.
The relation R is said to be transitive if $\left( {x,y} \right) \in R {\text{ and }} \left( {y,z} \right) \in R \Rightarrow \left( {x,z} \right) \in R {\text{ for }}x, y, z \in A$.
As the given set A contains three elements, given as A = \left\\{ {x,y,z} \right\\} and the relation $R$ does not contain $\left( {z,z} \right)$ and $z \in A$, so the relation $R$ is not reflexive.
As $\left( {z,x} \right)$ belongs to the given relation $R$ and $\left( {x,z} \right)$ does not belong to the given relation $R$. So by using the definition of symmetry.
So, it can be concluded that the given relation is not symmetric.
As $\left( {z,x} \right) \& \left( {x,x} \right)$ both belong to the given relation $R$ and $\left( {z,x} \right)$ also belong to the given relation$R$ and $\left( {z,y} \right) \& \left( {y,y} \right)$ both belong to the given relation $R$ and $\left( {z,y} \right)$also belong to the relation $R$.
From the above argument it can be concluded that, if $\left( {x,y} \right)\& \left( {y,z} \right)$ are belong to $R$ implies that $\left( {x,z} \right)$ belongs to the relation $R$, then the relation is transitive.
Therefore, the given relation is transitive.

Hence, the correct option is B.

Note:
A relation is a relationship between sets of values or it is a subset of the Cartesian product. A function is a relation in which there is only one output for each input and a relation is denoted by $R$ and a function is denoted by $F$.