Relations and Functions

Introduction

In mathematics, the concept of relation and function helps us understand how two sets of objects are connected through specific rules or correspondences. These are fundamental ideas in algebra and calculus, widely used in computer science, economics, and data analysis.

Relations

A relation R from a non-empty set A to a non-empty set B is a subset of the Cartesian product A × B. That means a relation is any collection of ordered pairs (a, b), where a ∈ A and b ∈ B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.

The set of all first elements in a relation R, is called the domain of the relation R, and the set of all second elements called images, is called the range of R.

Mathematically:

If R ⊆ A×B, then R is a relation from A to B.

Example:

Let:

• Set A = {1,4,5} 

• Set B = {2,3} 

Let R= {(1,2), (4,3), (5,3)}

• Domain of R = Set of all first elements = {1,4,5} 

• Range of R = Set of all second elements = {2,3} 

Functions (Mappings)

A function is a special kind of relation in which each element of the domain is mapped to exactly one element in the range. In simple terms, a function gives a unique output for every input.

Mathematically:

Let f: A → B, then for every a ∈ A, there exists only one b ∈ B such that f(a) = b.

Example:

Let f(x) =2x and x ∈ {1,2,3,4}

Then:

• f(1) = 2 

• f(2) = 4 

• f(3) = 6 

• f(4) = 8 

Thus, the function is:

f = {(1,2), (2,4), (3,6), (4,8)}

• Domain = {1,2,3,4} 

• Range = {2,4,6,8} 

Key Takeaways

• A relation connects elements of two sets through ordered pairs.

• A function is a relation where each input has exactly one unique output.