Question

Let R be any relation in the set A of human beings in a town at a particular time. If R=\left\\{ \left( x,y \right):x\,is\,wife\,of\,y \right\\}, Then R is
(a) Reflexive
(b) Symmetric
(c) Transitive
(d) An equivalence relation

Answer 1

Hint: Think of the basic definition of the types of relations given in the figure and try to check whether the relation mentioned in the questions satisfies the condition for any type of relation or not.

Complete step-by-step solution -
A is a set of all human beings in a town. Given that R is a relation such that R=\left\\{ \left( x,y \right):x\,is\,wife\,of\,y \right\\}
Now, a relation is symmetric if for an ordered pair $\left( x,y \right)$ in a relation, ordered pairs $\left( y,x \right)$ also belong to that relation. Here $\left( x,y \right)$ belongs to relation R when x is wife of y. Now, if x is wife of y, then y will be husband of x and not wife of x. So $\left( y,x \right)$, is i.e. y is wife of x is not true. Hence $\left( y,x \right)$ does not belong to R when $\left( x,y \right)$ belongs to R.
Therefore, R is not symmetric.
Also, a relation is reflexive, when the ordered pair $\left( x,x \right)$ belongs to a relation for all elements x of a set.
Here, R will be reflexive, if for all humans in town, $\left( x,x \right)$ will belong to R. but $\left( x,x \right)$ belong to R means that x is wife of x and we know that a person cannot be wife of themselves. Hence, $\left( x,x \right)$ does not belong to R.
Therefore, R is not reflexive.
Also, a relation is transitive, when for two ordered pairs $\left( x,y \right)$ and $\left( y,z \right)$ which both belong to a relation, a third pair $\left( x,z \right)$ will also belong to a relation.
Here, if $\left( x,y \right)$ belong to R, that is x is wife of y, then y is male and hence y cannot be wife of z. So $\left( y,z \right)$ will not belong to R.
Hence transitivity cannot occur here. Therefore, R is not transitive.
Hence, R is not reflexive, not symmetric and also not transitive. Therefore, none of the options is true.

Note: In this type of question, when writing tabular form of relation is not possible, consider examples of the situation to solve the question. We proceed with the definitions of relation and check whether the relation follows the definition or not.