1. What a Relation Is
A relation connects elements of one set with elements of another set using ordered pairs. It shows how one value is linked to another. A relation does not need to connect every element — it simply picks certain pairs that satisfy some rule or idea.
2. Relation as a Subset of the Cartesian Product
If two sets A and B are given, then all possible ordered pairs form the Cartesian product \( A \times B \). A relation from A to B is any subset of this product. So a relation is made by selecting some (or none) of the pairs from \( A \times B \).
2.1. Definition
R \subseteq A \times B
2.2. Example
Let:
A = \{1,2,3\},\quad B = \{4,5\}
A \times B = \{(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)\}
A possible relation is:
R = \{(1,5),(3,4)\}
This relation connects 1 with 5, and 3 with 4.
3. Relations on a Single Set
Sometimes the relation is formed within the same set. This means the relation is a subset of \( A \times A \). These are called relations on a set and are widely used in symmetry, ordering, and equivalence concepts.
3.1. Example
A = \{a,b,c\}
A \times A = \{(a,a),(a,b),(a,c),(b,a),(b,b),(b,c),(c,a),(c,b),(c,c)\}
R = \{(a,b),(b,c)\}
This relation shows which elements are linked inside the same set.
4. Ways to Describe a Relation
A relation can be written in different forms depending on what feels clearer:
4.1. Listing Ordered Pairs
Directly writing the pairs, such as:
R = \{(1,2),(2,4),(3,6)\}
4.2. Set-Builder Form
Describing the rule behind the relation:
R = \{(x,y) \mid y = 2x \}
4.3. Verbal Form
Writing the idea in words, such as “y is twice x”.
5. Examples to Understand the Idea
- Relation “is less than” between numbers → pairs like (2,5), (3,7)
- Relation “has same colour as” between objects
- Relation “divides” between numbers → pairs like (2,6), (3,15)
6. Important Notes
Some simple points to remember:
6.1. Key Points
- A relation is a set of ordered pairs.
- It always comes from a Cartesian product.
- It may include few, many, or none of the possible pairs.
- Relations are the base for many ideas like functions, symmetry, and ordering.