A relation in mathematics is defined as the relationship between two sets.
- If we are given two sets, set A and set B, and set A has a relation with set B, then each value of set A is related to a value of set B through some unique relation.
- Here, set A is called the domain of the relation, and set B is called the range of the relation.
Example: Suppose there are two sets X = {4, 36, 49, 50} and Y = {1, -2, -6, -7, 7, 6, 2}.
A relation R states that "(x, y) is in the relation R if x is a square of y" can be represented using ordered pairs,
- R = {(4, -2), (4, 2), (36, -6), (36, 6), (49, -7), (49, 7)}
Also, the image added below shows two sets, A and B, and the relation between them.
- Set A = {x, y, z}
- Set B = {1, 2, 3}
Representation of Relations
In mathematics or set theory we can represent the relation using different techniques, and the two important ways to represent the set are,
1. Set Builder Notation
If a relation between two sets is represented using the logical formula, then this type of representation is called the set builder notation.
For example, if we are given two sets, set X = {2, 4, 6} and set Y = {4, 8, 12}. Then, on observing clearly, we can see that each element of set Y is twice each element of set X the relation between them is,
R {(a, b): b is twice of a, a ∈ X, b ∈ Y}
2. Roaster Form
Roaster form is another way of representing a relation. In roaster form, we use ordered pairs to represent the relation.
For example, if we are given two sets, set X = {2, 4, 6} and set Y = {4, 8, 12}. Then the relation between set X and set Y is represented using the relation R such that,
R = {(2, 4), (4, 8), (6, 12)}
Graphing Relations
Relations can be easily represented on the graphs, and representing them on graphs is an easy way of explaining them. The ordered pair in a relation represents a coordinate that can be plotted on the Cartesian coordinate system.
- Substitute x with random numerical values in the relation.
- Find the corresponding y value of the respective x value.
- Write the ordered pair such that, {(x, y)}
- Plot these points and join them to find the required curve.
The graph of the relation y = x2 is added below,
Importance in Computer Science
Relations are widely used in computer science, often without explicitly calling them “relations.” Some key applications include:
1. Databases (Relational Model)
- The term relational database comes directly from the mathematical concept of a relation.
- A database table is a set of tuples — exactly what a relation is in set theory.
2. Graphs and Networks
- A graph can be seen as a relation between vertices.
- In an undirected graph, the relation is symmetric.
- In a directed graph, the relation is not necessarily symmetric.
3. State Machines
- In automata theory, the transition function is a relation between states and inputs to next states.
4. Other Applications
- Data Structures: Adjacency matrices and adjacency lists in graphs are ways to store relations between nodes.
- Information Retrieval: Search engines model the relation between documents and keywords.
➢Practice: Solved Examples
