We have been given that,

A is the set of all human beings in a town at a particular time.

Here, R is the binary relation on set A.

So, recall that

R is reflexive if for all x ∈ A, xRx.

R is symmetric if for all x, y ∈ A, if xRy, then yRx.

R is transitive if for all x, y, z ∈ A, if xRy and yRz, then xRz.

Using these criteria, we can solve these.

We have,

R = {(x, y): x and y live in the same locality}

Check for Reflexivity:

Since x & x are the same people, then, x & x live in the same locality.

Take yourself, for example, if you lived in colony x then you live in colony x.

Since you can’t live in two places at a particular time.

So, ∀ x ∈ A, then (x, x) ∈ R.

∴ R is Reflexive.

Check for Symmetry:

If x & y live in the same locality, then, y & x also lives in the the same locality.

If you & your friend, Chris are neighbors, then you and Chris are neighbors only.

The only difference is in the way of writing, either you write your name and your friend’s name or your friend’s name and your name, it’s the same.

So, if (x, y) ∈ R, then (y, x) ∈ R.

∀ x, y ∈ R

∴ R is Symmetric.

Check for Transitivity:

If x & y lives in the same locality and y & z lives in the same locality.

Then, x & z also lives in the same locality.

Say, if she and I were living in colony x and she & you were also working in colony x. Then, you and I are living in the same colony.

So, if (x, y) ∈ R and (y, z) ∈ R, then (x, z) ∈ R.

∀ x, y, z ∈ A

∴ R is Transitive.

Hence, R is reflexive, symmetric and transitive.