1. What a Relation Means
A relation connects elements of one set with elements of another set using ordered pairs. It shows how one value is linked or associated with another. The idea comes from the Cartesian product, where all possible pairs are listed, and a relation picks some of those pairs.
This makes relations useful for describing patterns, links, or rules between sets.
2. Relations Come from Sets
If two sets A and B are given, then every combination of one element from A and one from B forms the Cartesian product. A relation chooses certain pairs from this product. So a relation is simply a subset of the Cartesian product.
2.1. Example
Let:
A = \{1,2,3\}, \quad B = \{4,5\}
The Cartesian product is:
A \times B = \{(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)\}
A relation could be:
R = \{(1,4),(2,5)\}
This relation shows that 1 is linked to 4, and 2 is linked to 5.
3. How to Think About a Relation
A relation can be viewed in many simple and intuitive ways. It can describe matching, comparison, or connection between two sets. The same idea works even when both sets are the same.
3.1. Examples
- "is greater than" between numbers → pairs like (5,3), (10,2)
- "is a friend of" between people → pairs like (A,B)
- "has the same remainder when divided by 3" → pairs like (4,1), (7,4)
4. Relations Using Ordered Pairs
Ordered pairs are the basic building blocks of a relation. In each ordered pair:
- The first element comes from the first set.
- The second element comes from the second set.
The pairs show exactly how values are connected.
4.1. Example
Let:
A = \{a,b,c\}
A relation on A could be:
R = \{(a,a),(a,c),(b,c)\}
This tells us which elements are related, one pair at a time.
5. Why Relations Are Useful
Relations help describe connections or rules between elements. They appear in mappings, comparisons, grouping, and many logical or mathematical ideas. Once a relation is written using ordered pairs, it becomes easy to understand the pattern hidden in the data.