Ch 4. Relations
4.4 Domain, Range, Field, and Graphical Representation of Relations
The domain and range describe the sets of elements that participate in a relation. The domain of a relation is the set of all its first elements. Given a relation [latex]R \subseteq A \times B[/latex], we write the domain of [latex]R[/latex] as
[latex]\text{Dom}(R) = \{ a \in A \mid \exists b \in B \text{ such that } (a, b) \in R \}[/latex]
The range of a relation is the set of all its second elements. We write the range of [latex]R[/latex] as
[latex]\text{Ran}(R) = \{b \in B \mid \exists a \in A \text{ such that } (a, b) \in R \}[/latex]
Example 4.62
Let [latex]R = \{ (1, 2), (2, 3), (3, 4) \}[/latex]. Then [latex]\text{Dom}(R) = \{ 1, 2, 3 \}[/latex] and [latex]\text{Ran}(R) = \{ 2, 3, 4 \}[/latex].
The field of a relation is the set of all elements that appear in any position in the ordered pairs of the relation. We write the field of [latex]R[/latex] as
[latex]\text{Field}(R) = \text{Dom}(R) \cup \text{Ran}(R)[/latex]
Example 4.63
Using the same relation as Example 4.62, we have [latex]\text{Field}(R) = \{ 1, 2, 3, 4 \}[/latex].
As discussed in Section 4.2, relations can be visualized using directed graphs. In a directed graph, each element of the set is represented as a vertex, and each ordered pair [latex](a, b) \in R[/latex] is represented as a directed edge from vertex [latex]a[/latex] to vertex [latex]b[/latex]. The visual representation helps identify properties such as reflexivity (loops), symmetry (bidirectional edges), and transitivity (paths).
Example 4.64
Let [latex]A = \{ 1, 2, 3 \}[/latex] and [latex]R = \{ (1, 2), (2, 3), (3, 1) \}[/latex]. The directed graph is shown in the figure below.
This forms a cycle, showing that the relation is not reflexive (no loops), not symmetric (no reverse edges), but transitive in a circular sense.
Real-World Example 4.4: Social Media Mentions
If you are a user of a social media platform, you can mention other users in your posts and other users can also mention you. Let
[latex]U[/latex] = {Aiden, Benjamin, Chloe, Daniel}
Define the relation [latex]M \subseteq U \times U[/latex] where [latex](a, b) \in M[/latex] means user [latex]a[/latex] mentioned user [latex]b[/latex]. Suppose
[latex]M[/latex] = {(Aiden, Benjamin), (Benjamin, Chloe), (Chloe, Daniel)}
Then
[latex]\text{Dom}(M)[/latex] = {Aiden, Benjamin, Chloe}
[latex]\text{Ran}(M)[/latex] = {Benjamin, Chloe, Aiden}
[latex]\text{Field}(M)[/latex] = {Aiden, Benjamin, Chloe}
The directed graph in the figure below shows a cycle of mentions among three users.
Daniel is not involved in any mention, so he does not appear in the domain, range, or field.
