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 work at the same place}

Check for Reflexivity:

Since x & x are the same people then, x & x works at the same place.

Take yourself, for example, if you work at Bloomingdale then you work at Bloomingdale.

Since you can’t work in two places at a particular time,

So, ∀ x ∈ A, then (x, x) ∈ R.

∴ R is Reflexive.

Check for Symmetry:

If x & y works at the same place, then, y and x also work at the same place.

If you & your friend, Chris was working in Bloomindale, then Chris and you are working in Bloomingdale 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 ∈ A

∴ R is Symmetric.

Check for Transitivity:

If x & y works at the same place and y & z works at the same place.

Then, x & z also works at the same place.

Say, if she & I was working in Bloomingdale and she & you were also working in Bloomingdale. Then, you and I are working in the same company.

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.