1. What a Digraph Representation Means
A digraph (directed graph) shows a relation using points (called nodes) and arrows (called directed edges). Each element of the set becomes a node, and every ordered pair (a,b) in the relation is drawn as an arrow from a to b.
This gives a clear picture of how elements connect.
2. How a Digraph Is Built
To draw a digraph for a relation R on a set A:
- Draw a node for every element of A.
- For each ordered pair (a,b) in R, draw an arrow from node a to node b.
- Self-pairs (a,a) are shown as loops at node a.
3. Example of a Digraph
Let A = {1,2,3} and:
R = \{(1,2),(2,3),(1,3),(3,3)\}
Constructing the digraph:
3.1. Visual Description
- Arrow 1 → 2
- Arrow 2 → 3
- Arrow 1 → 3
- Loop at 3 (because of (3,3))
The digraph shows how the relation moves from one element to another.
4. Showing Self-Pairs
A self-pair appears when an element relates to itself. In a digraph, this is drawn as a curved loop starting and ending at the same node.
4.1. Example
R = \{(a,a),(a,b)\}
This has a loop at a and an arrow from a to b.
5. Using Digraphs to Observe Properties
Digraphs make it easy to spot important relation properties:
5.1. Reflexive
Every node must have a loop.
5.2. Symmetric
Every arrow a → b must have a reverse arrow b → a.
5.3. Antisymmetric
If both arrows a → b and b → a exist, then a and b must be the same node.
5.4. Transitive
If there is a → b and b → c, then a → c should also appear.
6. Example: Identifying Properties Using a Digraph
Let A = {x,y,z} and R = {(x,y),(y,x),(y,z)}.
Digraph picture:
- Arrows x ↔ y (two-way)
- Arrow y → z
6.1. Observations
- Symmetric between x and y.
- No loop at any node → not reflexive.
- x → y and y → z exist, but x → z is missing → not transitive.
7. Why Digraphs Are Useful
Digraphs give a quick visual understanding of how a relation behaves. They help identify patterns, check properties, and understand structure without scanning long lists of ordered pairs.
This makes them especially helpful for relations defined on small sets.