Question

Let A be the set of all human beings in a town to a particular time. Determine whether the relation $R=\left\{ \left( x,y \right)x\,\,is\,wife\,of\,y \right\}$ is reflexive, symmetric and transitive.

Answer 1

Hint:Here, we will use the definitions of reflexive, symmetric and transitive relations to check whether the given relations are reflexive, symmetric or transitive.

Complete step-by-step answer:
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, an ordered pair $\left( y,x \right)$ also belongs 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, $\left( y,x \right)$ i.e. y is the 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 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 belongs to a relation, a third pair $\left( x,z \right)$ will also belong to a relation.
Here, if $\left( x,y \right)$ belongs 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.

Note: Students should note here that for a relation to be a particular type of relation, i.e. reflexive, symmetric or transitive, all its elements must satisfy the required conditions. If we are able to find even a single counter example, i.e. if any of the elements doesn’t satisfy the criteria, then we can’t proceed further.