Question

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

Answer 1

In the given question, we are given a relation on the set of all human beings in a town at a particular time. We have to find whether the given relation is reflexive, symmetric and transitive. Relation R is satisfied if both the elements of the sets belong to the same locality. We will look for the reflexivity, symmetric and transitivity conditions of the relation one by one by introducing some variables such as x, y and z that belong to the set A of all human beings.

Complete step by step answer:
In the problem, we are given the relation between x and y such that x and y live in the same locality. So, R = \left\\{ {\left( {x,y} \right):\,\,{\text{x}}\,{\text{and y lives in the same locality}}} \right\\}. Now, we have to find if the function provided to us in the question is reflexive, symmetric and transitive. Reflexive relation is a type of relation in which each element maps to itself. So, for a relation to be reflexive, $\left( {x,x} \right)$ should belong to the relation for every value of x.

For relation R to be reflexive, $\left( {x,x} \right)$ should belong to R. We know that x and x belong to the same locality. Hence, $\left( {x,x} \right)$ belongs to the relation R. Therefore, relation R is reflexive. Symmetric function is a function in which an element maps to another element and the converse also holds true. So, for a relation to be reflexive, $\left( {x,y} \right)$ and $\left( {y,x} \right)$ should both belong to the relation for every value of x.

For relation R to be reflexive, $\left( {x,y} \right)$ and $\left( {y,x} \right)$ should belong to R. We know that if x and y belong to the same locality. Then, y and x also belong to the same locality. Hence, $\left( {x,y} \right)$ and $\left( {y,x} \right)$ belong to the relation R. So, the relation R is symmetric.

A function is said to be a transitive function if an element maps to another element, that other element relates to a third element and the first element also maps to the third element. So, for a relation to be reflexive, $\left( {x,y} \right)$, $\left( {y,z} \right)$ and $\left( {z,x} \right)$ all belong to the relation for every value of x.

We know that if x and y belong to the same locality, y and z belong to the same locality.Then, y and x also belong to the same locality. Hence, $\left( {x,y} \right)$, $\left( {y,z} \right)$ and $\left( {z,x} \right)$ belongs to the relation R. So, the relation R is transitive.

Therefore, the relation R is reflexive, symmetric and transitive.

Note: We must know the different types of relations in order to solve the given problem. One must know the definitions of reflexive, symmetric and transitive relations in order to classify the relations. One should also know that if a relation is reflexive, symmetric and transitive, then the relation is called an equivalent relation.