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 is exactly 7 cm taller than y}

Answer 1

(i) Reflexive:

R = {(x, y) : x is exactly 7 cm taller than y}

Now, (x, x) ∉ R

Since a human being (x) cannot be taller than himself.

∴ R is not reflexive.

(ii) Symmetric:

Now, let (x, y) ∈ R

⇒ x is exactly 7 cm taller than y.

Then, y is not taller than x.

∴ (y, x) ∉ R

Indeed, if x is exactly 7 cm taller than y, then y is exactly 7 cm shorter than x.

∴ R is not symmetric.

(iii) Transitive:

Now, let (x, y), (y, z) ∈ R

⇒ x is exactly 7 cm taller than y, and y is exactly 7 cm taller than z.

⇒ x is exactly 14 cm taller than z.

∴ (x, z) ∉ R

∴ R is not transitive.

Hence, R is not reflexive, not symmetric and not transitive.