Answer :

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 is father of y}

**Check for Reflexivity:**

Since x and x are the same people.

Then, x cannot be the father of itself.

A person cannot be a father of itself.

Leo is the father of Thiago

So, ∀ x ∈ A, then (x, x) ∉ R.

**∴** **R is not reflexive.**

**Check for Symmetry:**

If x is the father of y.

Then, y cannot be the father of x.

If Sam is the father of Mac, then Mac is the son of Sam.

Mac cannot be the father of Sam.

So, if (x, y) ∈ R, then (y, x) ∉ R.

∀ x, y ∈ A

**∴** **R is not symmetric.**

**Check for Transitivity:**

If x is the father of y and y is the father of z, then, x is not the father of z.

Take Mickey, Sam, and Mac.

If Mickey is the father of Sam, and Sam is the father of Mac.

Thus, Mickey is not the father of Mac, but the grandfather of Mac.

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

∀ x, y, z ∈ A

**∴** **R is not transitive.**

**Hence, R is neither reflexive, nor symmetric, nor transitive.**

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