How to Graph a Relation and Identify Its Properties

A relation on a set can be represented as a directed graph, which makes its structure easier to inspect. Each element becomes a vertex, and each ordered pair in the relation becomes an arrow. From the graph, you can check common properties such as reflexivity, symmetry, antisymmetry, and transitivity.

This method applies most directly when the set is finite. For an infinite set, the same definitions still apply, but drawing every vertex and arrow may not be possible.

Turn the relation into a directed graph

A relation $R$ on a set $A$ is a set of ordered pairs whose entries come from $A$. The notation $aRb$ means $(a,b)\in R$.

To graph a finite relation:

1. Draw one vertex for each element of $A$.
2. For every pair $(a,b)\in R$, draw an arrow from $a$ to $b$.
3. Draw a loop at $a$ if $(a,a)\in R$.

The direction matters: an arrow from $a$ to $b$ does not automatically mean there is an arrow from $b$ to $a$. For instance, a relation written as “is less than or equal to” is represented by arrows from smaller elements to larger ones.

Read the standard properties from the graph

The definitions below describe relations on a set $A$. They also give practical checks for a finite directed graph.

Property	Definition	Graph test
Reflexive	$aRa$ for every $a\in A$	Every vertex has a loop.
Symmetric	If $aRb$, then $bRa$	Every arrow between distinct vertices has a reverse arrow.
Antisymmetric	If $aRb$ and $bRa$, then $a=b$	No two distinct vertices have arrows in both directions.
Transitive	If $aRb$ and $bRc$, then $aRc$	Every two-edge path $a\to b\to c$ has a direct arrow $a\to c$.

Symmetry and antisymmetry are different tests. A symmetric relation requires reverse arrows; an antisymmetric relation rules out reverse arrows between distinct vertices. A relation can be both (for example, equality), neither, or just one of the two.

When checking transitivity, consider every possible two-edge path, including paths that use loops. The required direct arrow may be a loop when the path begins and ends at the same vertex.

Worked example

Let $A=\{1,2,3\}$ and define

$$R=\{(1,1),(2,2),(3,3),(1,2),(2,1),(2,3)\}.$$

The graph has vertices $1,2,3$, a loop at each vertex, arrows in both directions between $1$ and $2$, and an arrow from $2$ to $3. There is no arrow from$3$to$2$, and no arrow from$1$to$3$.

Now test the properties:

• Reflexive: Yes. All three loops are present: $(1,1)$, $(2,2)$, and $(3,3)$.
• Symmetric: No. The arrow $2\to3$ has no reverse arrow $3\to2$.
• Antisymmetric: No. There are arrows $1\to2$ and $2\to1$, even though $1\ne2$.
• Transitive: No. The path $1\to2\to3$ would require an arrow $1\to3$, which is absent.

One failed test is enough to show that a relation does not have a property. You do not need to check every pair after finding a counterexample, although checking all pairs can help confirm the properties that do hold.

A useful comparison: the order relation

On $A=\{1,2,3\}$, consider the relation $\leq$. Its pairs are

$$\{(1,1),(1,2),(1,3),(2,2),(2,3),(3,3)\}.$$

The graph has a loop at every vertex, and arrows from each number to itself and to every larger number. This relation is reflexive, antisymmetric, and transitive, but not symmetric: $1\leq2$ while $2\nleq1$. A relation that is reflexive, antisymmetric, and transitive is called a partial order. The graph can be drawn more simply as a Hasse diagram by omitting loops and arrows implied by transitivity, but those omissions are a drawing convention; the relation itself still contains the corresponding pairs.

Common mistakes

• Ignoring direction: For a relation, $aRb$ and $bRa$ are separate facts. Draw separate arrows unless the graph is explicitly being used to represent an undirected relation.
• Forgetting loops: Reflexivity requires a loop at every element, not merely at some of them. A loop is also a pair that can matter when testing transitivity.
• Confusing antisymmetry with asymmetry: Antisymmetry allows loops and allows arrows in one direction between distinct elements. It only forbids arrows in both directions between distinct elements. Asymmetry is stronger: it forbids loops as well as reverse-arrow pairs.
• Checking only adjacent pairs for transitivity: Transitivity concerns every two-edge path. If $a\to b$ and $b\to c$, check whether $a\to c$ is present, even when the three vertices are different.
• Treating a missing arrow as proof of a property: A missing reverse arrow can support antisymmetry, but symmetry requires every arrow to have a reverse. Always match the graph to the exact definition.

For a large finite relation, you can also record the pairs in a matrix: put a $1$ in row $a$, column $b$ exactly when $aRb$. The same tests apply, but the directed graph is often easier to interpret visually.