Question

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

Answer 1

Reflexive is if (a, a) $\in$R for every a $\in$ A. Symmetric id if (a, b) $\in$ R and then (b, a) $\in$ R. Transitive is if (a, b) $\in$ R and (b, c) $\in$ R. Check with conditions with the given relation R, for the variable x and y where they live in the same locality.

Complete step-by-step answer:
Before we get to the question, let us discuss what is reflexive, symmetric and transitive relation is.
For a relation R on a set A is called reflexive if (a, a) $\in$ R for every element a $\in$ A. A relation R on a set A is called symmetric if (b, a) $\in$ R whenever (a, b) $\in$ R, for all a, b $\in$ A. Now a relation is transitive, if (a, b) $\in$ R and (b, c) $\in$ R, then (a, c) $\in$ R.
Thus, if a relation is reflexive, symmetric and transition, then it is an equivalence relation.
Now, given to us, R = {(x, y): x and y live in the same locality}
A is the set of all human beings in a town at a particular time.
Let us first check if the relation is symmetric.
Let (x, y) $\in$ R, x and y live at the same locality, then (y, x) $\in$ R. So, it is a symmetric relation.
Now let us check transitivity.
Let (x, y) $\in$ R and (y, z) $\in$ R. Then, x and y live at the same locality, y and z also live in the same locality. Then x, y and z all live at the same locality. Thus x and z live at the same locality. i.e. (x, z) $\in$ R. So, R is a transitive relation.
Hence from this we can say that R is reflexive, symmetric and transitive.
Thus function R is an equivalence relation.

Note: As a consequence of the reflexive, symmetric and transitive property, any equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class.