Section 1.1 & 1.2: Types of Relations

In this section, we would like to study different types of relations.

Definition: Empty Relation
A relation \(R\) in a set \(A\) is called empty relation, if no element of \(A\) is related to any element of \(A\), i.e., \(R = \phi \subseteq A \times A\).

For illustration, consider a relation \(R\) in the set \(A = \{1, 2, 3, 4\}\) given by

\[ R = \{(a, b) : a - b = 10\} \]

This is the empty set, as no pair \((a, b)\) satisfies the condition \(a - b = 10\).

Definition: Universal
A relation \(R\) in a set \(A\) is called universal relation, if each element of \(A\) is related to every element of \(A\), i.e., \(R = A \times A\).

Example: \[ R' = \{(a, b) : |a - b| \geq 0\} \]

is the whole set \(A \times A\), as all pairs \((a, b)\) in \(A \times A\) satisfy \(|a - b| \geq 0\).

Both the empty relation and the universal relation are sometimes called trivial relations.