Relation and functions | Part -2

Understanding Relations Through Ordered Pairs: Domain and Range

In mathematics, a relation refers to a set comprising ordered pairs. Each ordered pair consists of two elements: the first element represents the input (or domain), and the second element represents the output (or range). Relations help us understand how elements from one set correspond to elements in another.

Definition of a Relation

A relation is defined as a subset of the Cartesian product of two sets. If we have two sets A and B, then the Cartesian product is written as A × B and contains all possible ordered pairs (a, b) where a ∈ A and b ∈ B.

For example, consider the relation:

  R = {(1, 4), (2, 5), (3, 6), (2, 7)}
  

Each ordered pair indicates a connection between an element of the first set and one from the second set. Unlike functions, a relation may allow a single input to correspond to more than one output.

Finding the Domain of a Relation

The domain of a relation refers to the complete set of first components (input values) found in its ordered pairs. To determine the domain, list each unique first component:

  Domain of R = {1, 2, 3}
  

Finding the Range of a Relation

The range comprises all the second components (output values) from the given ordered pairs. To find it, collect all the second components without repetition:

  Range of R = {4, 5, 6, 7}
  

Key Points

• A relation can have multiple outputs for the same input (unlike a function).
• Relations are useful in various fields such as computer science, economics, and logic.
• To identify the domain and range, examine each pair and collect the first and second components respectively.

By interpreting a relation as a set of ordered pairs, one can easily determine which elements belong to the domain and which to the range. This foundational concept plays a crucial role in the study of functions and further mathematical analysis.