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

Hint:
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 \[F\].