2.1: Definitions
Let \(X\) and \(Y\) be sets. A relation from \(X\) to \(Y\) is a subset of \(X \times Y\) .
Alternatively, any set of ordered pairs is a relation. If \(Y=X\) , we say that \(R\) is a relation on \(X\) .
Notation. \(x\) Ry Let \(X\) and \(Y\) be sets and \(R\) be a relation on \(X \times Y\) . If \(x \in X\) and \(y \in Y\) , then we may express that \(x\) bears relation \(R\) to \(y\) (that is \((x, y) \in R\) ) by writing \(x R y\) .
So for \(X\) and \(Y\) sets, \(x \in X, y \in Y\) , and \(R\) a relation on \(X \times Y\) , \(x R y\) if and only if \((x, y) \in R\) .
Let \(\leq\) be the usual ordering on \(\mathbb{Q}\) . Then \(\leq\) is a relation on \(\mathbb{Q}\) . We write \[1 / 2 \leq 2\] to express that \(1 / 2\) bears the relation \(\leq\) to 2 .
Define a relation \(R\) from \(\mathbb{Z}\) to \(\mathbb{R}\) by \(x R y\) if \(x>y+3\) . Then we could write \(7 R \sqrt{2}\) or \((7, \sqrt{2}) \in R\) to say that \((7, \sqrt{2})\) is in the relation.
Let \(X=\{2,7,17,27,35,72\}\) . Define a relation \(R\) by \(x R y\) if \(x \neq y\) and \(x\) and \(y\) have a digit in common. Then \(R=\{(2,27),(2,72),(7,17),(7,27),(7,72),(17,7),(17,27),(17,72),\) , \((27,2),(27,7),(27,17),(27,72),(72,2),(72,7),(72,17),(72,27)\}\) . ExAMPLE 2.4. Let \(P\) be the set of all polygons in the plane. Define a relation \(E\) by saying \((x, y) \in E\) if \(x\) and \(y\) have the same number of sides.
How do mathematicians use relations? A relation on a set can be used to impose structure. In Example 2.1, the usual ordering relation \(\leq\) on \(\mathbb{Q}\) allows us to think of rational numbers as lying on a number line, which provides additional insight into rational numbers. In Example 2.4, we can use the relation to break polygons up into the sets of triangles, quadrilaterals, pentagons, etc.
A function \(f: X \rightarrow Y\) can be thought of as a very special sort of relation from \(X\) to \(Y\) . Indeed, the graph of the function is a set of ordered pairs in \(X \times Y\) , but it has the additional property that every \(x\) in \(X\) occurs exactly once as a first element of a pair in the relation. As we discussed in Section 1.3, functions are a useful way to relate sets.
Let \(X\) be a set, and \(R\) a relation on \(X\) . Here are some important properties the relation may or may not have.
\(R\) is reflexive if for every \(x \in X\) , \[x R x .\] Symmetric \(R\) is symmetric if for any \(x, y \in X\) , \[x R y \text { implies } y R x \text {. }\] Antisymmetric \(R\) is antisymmetric if for any \(x, y \in X\) , \[[(x, y) \in R \text { and }(y, x) \in R] \text { implies } x=y \text {. }\] Transitive \(R\) is transitive if for any \(x, y, z \in X\) , \[[x R y \text { and } y R z] \text { implies }[x R z] \text {. }\] Which of these four properties apply to the relations given in Examples 2.1-2.4 (Exercise 2.1)?