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Beginning Activity Beginning Activity 1: Properties of Relations
A4US

In previous mathematics courses, we have worked with the equality relation. For example, let \(R\) be the relation on \(\Z\) defined as follows: For all \(a, b \in \Z\text{,}\) \(a \mathrel{R} b\) if and only if \(a = b\text{.}\) We know this equality relation on \(\Z\) has the following properties:

  • For each \(a \in \Z\text{,}\) \(a = a\) and so \(a \mathrel{R} a\text{.}\)

  • For all \(a, b \in \Z\text{,}\) if \(a = b\text{,}\) then \(b = a\text{.}\) That is, if \(a \mathrel{R} b\text{,}\) then \(b \mathrel{R} a\text{.}\)

  • For all \(a, b, c \in \Z\text{,}\) if \(a = b\) and \(b = c\text{,}\) then \(a = c\text{.}\) That is, if \(a \mathrel{R} b\) and \(b \mathrel{R} c\text{,}\) then \(a \mathrel{R} c\text{.}\)

In mathematics, when something satisfies certain properties, we often ask if other things satisfy the same properties. Before investigating this, we will give names to these properties.

Definition.

Let \(A\) be a nonempty set and let \(R\) be a relation on \(A\text{.}\)

  • The relation \(R\) is reflexive on \(\boldsymbol{A}\) provided that for each \(x \in A\text{,}\) \(x \mathrel{R} x\) or, equivalently, \(\left( {x, x} \right) \in R\text{.}\)

  • The relation \(R\) is symmetric provided that for every \(x, y \in A\text{,}\) if \(x \mathrel{R} y\text{,}\) then \(y \mathrel{R} x\) or, equivalently, for every \(x, y \in A\text{,}\) if \(\left( {x, y} \right) \in R\text{,}\) then \(\left( {y, x} \right) \in R\text{.}\)

  • The relation \(R\) is transitive provided that for every \(x, y, z \in A\text{,}\) if \(x \mathrel{R} y\) and \(y \mathrel{R} z\text{,}\) then \(x \mathrel{R} z\) or, equivalently, for every \(x, y, z \in A\text{,}\) if \(\left( {x, y} \right) \in R\) and \(\left( {y, z} \right) \in R\text{,}\) then \(\left( {x, z} \right) \in R\text{.}\)

Before exploring examples, for each of these properties, it is a good idea to understand what it means to say that a relation does not satisfy the property. So let \(A\) be a nonempty set and let \(R\) be a relation on \(A\text{.}\)

1.

Carefully explain what it means to say that the relation \(R\) is not reflexive on the set \(A\text{.}\)

2.

Carefully explain what it means to say that the relation \(R\) is not symmetric.

3.

Carefully explain what it means to say that the relation \(R\) is not transitive.

To illustrate these properties, we let \(A = \left\{ {1, 2, 3, 4} \right\}\) and define the relations \(R\) and \(T\) on \(A\) as follows:

\begin{align*} R \amp = \left\{ {( {1, 1} ), ( {2, 2} ), ( {3, 3} ), ( {4, 4} ), ( {1, 3} ), ( {3, 2} )} \right\}\\ T \amp = \left\{ {( {1, 1} ), ( {1, 4} ), ( {2, 4} ), ( {4, 1} ), ( {4, 2} )} \right\} \end{align*}

4.

Draw a directed graph for the relation \(R\text{.}\) Then explain why the relation \(R\) is reflexive on \(A\text{,}\) is not symmetric, and is not transitive.

5.

Draw a directed graph for the relation \(T\text{.}\) Is the relation \(T\) reflexive on \(A\text{?}\) Is the relation \(T\) symmetric? Is the relation \(T\) transitive? Explain.