Empty and Universal Relation

Empty Relation

If Relation has no elements,

it is called empty relation

We write R = ∅

Universal Relation

If relation has all the elements,

it is a universal relation

Let us take an example

Let A = Set of all students in a girls school.

We define relation R on set A as

R = {(a, b): a and b are brothers}

R’ = {(a, b): height of a & b is greater than 10 cm}

Now,

R

R = {(a, b): a and b are brothers}

It is a girls school, so there are no boys in the school.

Hence, there cannot be a brother.

∴ R has no elements

⇒ R = ∅ i.e. R is an empty relation

R'

R’ = {(a, b): height of both a & b is greater than 10 cm}

Height of children are always greater than 10 cm (0.3 foot)

Thus, height of a and b will always be greater than 10 cm

∴ R has all the students of the school.

⇒ R is a universal relation, since it has all the elements

What are trivial relations?

Sometimes, empty relation and universal relation are called trivial relations