1. Why Relations Need Different Representations
A relation can be shown in many forms. Sometimes listing pairs is enough, but diagrams or tables can make the structure clearer. Each representation carries the same information — just presented differently.
2. Listing Ordered Pairs
This is the most direct way to show a relation. Every pair in the relation is written explicitly.
2.1. Example
A = \{1,2,3\},\ B = \{4,5\}
R = \{(1,4),(2,5)\}
This form clearly shows which elements are linked.
3. Arrow Diagram
An arrow diagram shows two sets as two groups of elements with arrows drawn from elements of the first set to elements of the second set. Each arrow represents one ordered pair in the relation.
3.1. Example
For the relation:
R = \{(1,4),(2,5)\}
We draw arrows:
- 1 → 4
- 2 → 5
This gives a simple visual picture of how elements are connected.
4. Table Representation
A table can be used to show a relation by marking each ordered pair with a symbol (usually a dot or 1). Rows represent elements of the first set, and columns represent elements of the second set.
4.1. Example
Relation:
R = \{(1,4),(2,5)\}
The table:
| A ↓ / B → | 4 | 5 |
|---|---|---|
| 1 | 1 | 0 |
| 2 | 0 | 1 |
| 3 | 0 | 0 |
“1” marks pairs that belong to the relation.
5. Set-Builder Representation
In set-builder form, a relation is expressed using a rule or condition. This captures the idea behind the relation without listing every pair.
5.1. Example
Relation of numbers where the second number is double the first:
R = \{(x,y) \mid y = 2x\}
6. Verbal or Rule Description
Sometimes the clearest way to express a relation is using simple language that describes the connection.
6.1. Example
“y is twice x” or “a is related to b if a divides b”.
7. Using Venn-Style Visuals
Although not common for all relations, when the sets overlap or when the relation is within the same set, simple overlapping diagrams can help show connections.
7.1. Example
For a relation on A = {a,b,c}, arrows inside the same circle can show links like a → b, b → c.
8. Choosing the Best Representation
The form depends on what you want to understand:
- Ordered pairs → clear list
- Arrow diagram → visual connections
- Table → structured check of all possibilities
- Set-builder → highlights the underlying rule
- Verbal → easiest to explain in words