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Section 8.1 Relations

Many expressions in math describe a relationship between two objects. We call these relations, or more precisely binary relations, because they are relations between two objects.

Again, these relations can be defined in terms of set theory.

Subsection 8.1.1 Relations

Definition 8.1.1.

Let \(A\) be a set. A binary relation (or simply relation) on \(A\text{,}\) is a subset \(R\) of the Cartesian product \(A \times A\text{.}\) If \((x,y) \in R\text{,}\) we say that \(x\) is related to \(y\text{,}\) and we write \(x R y\text{.}\) Otherwise, \(x\) is not related to \(y\text{,}\) or \(x \cancel{R} y\text{.}\)

The notation to represent a general relation (\(R\) here) can vary. Sometimes, a tilde \(\sim\) is used, i.e. \(x \sim y\text{.}\)

Consider the set \(S\) of people in your class. Let \(R\) be a relation on \(S\) such that for \(x, y \in S\text{,}\) \(x R y\) if people \(x\) and \(y\) share the same first letter of their first name.

Let \(R\) be the relation such that \(x R y\) if person \(x\) is a sibling of person \(y\text{.}\)

Equality \(=\text{,}\) or inequality (greater than, less than, greater than or equal to, less than or equal to) \(> \, \lt \, \geq \, \leq\) are all relations on \(\mathbb{R}\) (or \(\mathbb{Q}, \mathbb{Z}, \mathbb{N}\)). More generally, given a set \(S\text{,}\) we can consider the equality of objects, given by the subset,

\begin{equation*} \set{(x,x): x \in S} \subseteq S \times S \end{equation*}

Then, for any \(x \in S\text{,}\) \(x = x\text{,}\) and if \(x\) and \(y\) are different elements of \(S\text{,}\) then \(x \neq y\text{.}\)

Divides \(\mid\) is a relation on a set of integers, where \(a \mid b\) if \(B\) is a multiple of \(A\text{,}\) i.e. there exists \(k \in \mathbb{Z}\) such that \(b = ka\text{.}\)

If \(S = \set{1,2,3,4} \subseteq \mathbb{Z}\text{,}\) then \(\mid\) is the set,

\begin{equation*} \mid \,= \set{(1,1),(1,2),(1,3),(1,4),(2,2),(2,4),(3,3),(4,4)} \end{equation*}

Slightly more generally, we can consider a relation between two different sets.

Definition 8.1.5. Relation between two different sets.

Let \(A, B\) be sets. A relation between \(A\) and \(B\text{,}\) \(R\text{,}\) is a subset of \(A \times B\text{.}\) If \(R = A \times B\) between \(A\) and \(B\text{,}\) then \(x R y\) for all \(x \in A\) and \(y \in B\text{.}\)

Subsection 8.1.2 Visualizing Relations

Let \(R\) be a relation on a set \(A\text{.}\) Draw each element in \(A\) as a point, and for any two elements \(x\) and \(y\text{,}\) if \(x R y\text{,}\) draw an arrow from \(x\) to \(y\text{.}\)