Definition: A relation between sets \(A\) and \(B\) is a subset of \(A \times B\). A relation on \(A\) is a subset of \(A^2.\)

If \(R \subset A \times B\) is a relation, instead of writing \((a, b) \in R\) we usually write \(aRb\). Also, instead of writing \((a, b) \not\in R\) we usually write \(a\not Rb\).

This example is unconventional, but gets across the idea behind relations.

Let \(A =\{\)one, two, three\(\},\) and let \(B =\{\)one plus one, one plus two\(\}.\) Then

\(A \times B = \{\)(one, one plus one), (one, one plus two), (two, one plus one), (two, one plus two), (three, one plus one), (three, one plus two)\(\}.\)

We will define \(R\) to be the relation that the word in \(A\) is equal to the expression in \(B.\)

\(R = \{\)(two, one plus one), (three, one plus two)\(\}\)

For elements \(a \in A\) and \(b \in B,\) instead of writing \((a, b) \in R\) write "\(a\) equals \(b\)". Instead of writing \((a, b) \not\in R,\) write "\(a\) does not equal \(b\)". So, since (two, one plus one) \(\in R,\) we write "two equals one plus one." Similarly, since (one, one plus two) \(\not\in R,\) we write "one does not equal one plus two."

Let \(A = \{1, 2, 3, 4\}.\) Then "less than" relation on \(A\) can be written \[< = \{(1,2), (1,3), (1,4), (2,3), (2,4), (3,4)\}.\] Instead of writing \((1,3) \in <\), we write \(1<3\). Similarly, instead of writing \((4,2) \not\in <\), we write \(4 \not < 2\).

The \(=\) relation on \(A\) is \[= = \{(1,1), (2,2), (3,3), (4,4)\}\] Since \(=\) is a relation, \((1, 1) \in =\) is written \(1 = 1\) and \((1, 2) \not\in =\) is written \(1 \neq 2.\)

Keep in mind that a relation between \(A\) and \(B\) is just a way we think about certain subsets of \(A \times B.\) Any subset of \(A \times B\) is a relation, but we think in terms of relations or subsets depending on the context.

For example, if \(A = \{1, 2, 3, 4\}\) and \(B = \{1, 2, 3, 4, 5\},\) then

\(\{(2, 2), (2, 5)\} \subset A \times B.\)

So, we could think of \(\{(2, 2), (2, 5)\}\) as a relation, but what kind of relationship is it? Without reason to consider this a relation, we continue to think in terms of sets and subsets.