Relations play a fundamental role in probability theory, as in most other areas of mathematics.
Suppose that \(S\) and \(T\) are sets. A relation from \(S\) to \(T\) is a subset of the product set \(S \times T\).
A relation from a set \(S\) to itself is a relation on \(S\).
As the name suggests, a relation \(R\) from \(S\) into \(T\) is supposed to define a relationship between the elements of \(S\) and the elements of \(T\), and so we usually use the more suggestive notation \(x\,R\,y\) when \((x, y) \in R\). Note that the domain of \(R\) is the projections of \( R \) onto \( S \) and the range of \( R \) is the projection of \( R \) onto \( T \).
Suppose that \(S\) is a set and recall that \(\mathscr{P}(S)\) denotes the power set of \(S\), the collection of all subsets of \(S\). Then membership \(\in\) is a relation from \(S\) to \(\mathscr{P}(S)\).
Membership is perhaps the most important and basic relationship in mathematics. Indeed, for us, it's a primitive (undefined) relationship—given \(x\) and \(A\), we assume that we understand the meaning of the statement \(x \in A\).
If \(S\) is a set then equality \(=\) is a relation on \(S\).
Equality is another primitive relation. That is, given \(x, \, y \in S\), we assume that we understand the meaning of the statement \(x = y\).
Other basic relations that we have seen are
These two belong to a special class of relations known as partial orders. Note that a function \(f\) from \(S\) into \(T\) is a special type of relation. To compare the two types of notation (relation and function), note that \(x \, f \, y\) means that \(y = f(x)\).
Since a relation is just a set of ordered pairs, the set operations can be used to build new relations from existing ones.
if \(Q\) and \(R\) are relations from \(S\) to \(T\), then so are \(Q \cup R\), \(Q \cap R\), \(Q \setminus R\).
If \(R\) is a relation from \(S\) to \(T\) and \(Q \subseteq R\), then \(Q\) is a relation from \(S\) to \(T\).
The restriction of a relation defines a new relation.
If \(R\) is a relation on \(S\) and \(A \subseteq S\) then \(R_A = R \cap (A \times A)\) is a relation on \(A\), called the restriction of \(R\) to \(A\).
The inverse of a relation also defines a new relation.
If \(R\) is a relation from \(S\) to \(T\), the inverse of \(R\) is the relation from \(T\) to \(S\) defined by \[ y\,R^{-1}\,x \text{ if and only if } x\,R\,y \]
Equivalently, \(R^{-1} = \{(y, x): (x, y) \in R\}\). Note that any function \(f\) from \(S\) into \(T\) has an inverse relation, but only when the \(f\) is one-to-one is the inverse relation also a function (the inverse function). Composition is another natural way to create new relations from existing ones.
Suppose that \(Q\) is a relation from \(S\) to \(T\) and that \(R\) is a relation from \(T\) to \(U\). The composition \(Q \circ R\) is the relation from \(S\) to \(U\) defined as follows: for \(x \in S\) and \(z \in U\), \(x(Q \circ R)z\) if and only if there exists \(y \in T\) such that \(x\,Q\,y\) and \(y\,R\,z\).
Note that the notation is inconsistent with the notation used for composition of functions, essentially because relations are read from left to right, while functions are read from right to left. Hopefully, the inconsistency will not cause confusion, since we will always use function notation for functions.
The important classes of relations that we will study in the next couple of sections are characterized by certain basic properties. Here are the definitions:
Suppose that \(R\) is a relation on \(S\).
The proofs of the following results are straightforward, so be sure to try them yourself before reading the ones in the text.
A relation \(R\) on \(S\) is reflexive if and only if the equality relation \(=\) on \(S\) is a subset of \(R\).
This follows from the definitions. \( R \) is reflexive if and only if \( (x, x) \in R \) for all \( x \in S \).
A relation \(R\) on \(S\) is symmetric if and only if \(R^{-1} = R\).
Suppose that \( R \) is symmetric. If \( (x, y) \in R \) then \( (y, x) \in R \) and hence \( (x, y) \in R^{-1} \). If \( (x, y) \in R^{-1} \) then \( (y, x) \in R \) and hence \( (x, y) \in R \). Thus \( R = R^{-1} \). Conversely, suppose \( R = R^{-1} \). If \( (x, y) \in R \) then \( (x, y) \in R^{-1} \) and hence \( (y, x) \in R \).
A relation \(R\) on \(S\) is transitive if and only if \(R \circ R \subseteq R\).
Suppose that \( R \) is transitive. If \( (x, z) \in R \circ R \) then there exists \( y \in S \) such that \( (x, y) \in R \) and \( (y, z) \in R \). But then \( (x, z) \in R \) by transitivity. Hence \( R \circ R \subseteq R \). Conversely, suppose that \( R \circ R \subseteq R \). If \( (x, y) \in R \) and \( (y, z) \in R \) then \( (x, z) \in R \circ R \) and hence \( (x, z) \in R \). Hence \( R \) is transitive.
A relation \(R\) on \(S\) is antisymmetric if and only if \(R \cap R^{-1}\) is a subset of the equality relation \(=\) on \(S\).
Restated, this result is that \( R \) is antisymmetric if and only if \( (x, y) \in R \cap R^{-1} \) implies \( x = y \). Thus suppose that \( R \) is antisymmetric. If \( (x, y) \in R \cap R^{-1} \) then \( (x, y) \in R \) and \( (x, y) \in R^{-1} \). But then \( (y, x) \in R \) so by antisymmetry, \( x = y \). Conversely suppose that \( (x, y) \in R \cap R^{-1} \) implies \( x = y \). If \( (x, y) \in R \) and \( (y, x) \in R \) then \( (x, y) \in R^{-1} \) and hence \( (x, y) \in R \cap R^{-1} \). Thus \( x = y \) so \( R \) is antisymmetric.
Suppose that \(Q\) and \(R\) are relations on \(S\). For each property below, if both \(Q\) and \(R\) have the property, then so does \(Q \cap R\).
Suppose that \(R\) is a relation on a set \(S\).
Suppose that \(R\) is a relation on a set \(S\).
Let \(R\) be the relation defined on \(\R\) by \(x\,R\,y\) if and only if \(\sin(x) = \sin(y)\). Determine if \(R\) has each of the following properties:
The relation \(R\) in is a member of the important class of equivalence relatons as is the equality relation \(=\) on a set \(S\) in
Let \(R\) be the relation defined on \(\R\) by \(x\,R\,y\) if and only if \(x^2 + y^2 \le 1\). Determine if \(R\) has each of the following properties: