We often want to be able to treat a relationship, such as the \(M\) (motherhood) relation discussed previously, as a mathematical object so we can reason about it.  Such an object is called a relation.  In most respects, relation is just the math word for relationship.

What kind of mathematical object is a relation?  One way to think about this is to as what makes one kind of relationship different from or the same as another.  There are basically two ways to answer this:

• Two relationships are the same if they mean the same thing.  This is called an intensional definition.
• Two relationships are the same if they are true of the same pairs of objects.  This is called an extensional definition.

The first of these is extremely complicated to reason about and so is almost never used in math, although it's important in philosophy, linguistics, and logic.

The second works for most mathematics and will work for everything we do here.¹ It's the definition used in set theory.  The set of pairs of objects a relationship is true of is called its extension.  In set theory, two relationships are the same if they have the same extension.  And the relation object is just the extension.  So relations are sets of pairs.  The motherhood relation would be defined as:

This means that when we type \(mMc\), it's really just a shorthand for \((m,c)\in M\).

Relations and the Cartesian product

Again, relations in set theory are just sets of pairs.  So any relation \(R\) between two sets \(A\) and \(B\) is going to be some set of pairs whose first element comes from \(A\) and whose second comes from \(B\).  That means \(R\) is a subset of the Cartesian product of \(A\) and \(B\):

As a result, it's very common for writers to say things like “let \(R\subseteq A\times B\) be a relation” rather than the more verbose "let \(R\) be a relation between elements of \(A\) and elements of \(B\)".

Notes