Functional Relation

We call a relation functional if every element in the starting set connects to at most one element in the target set. In other words: there is no case where the same starting element connects to two different outputs.

This is exactly the property that characterizes functions: each input corresponds to at most one output unambiguously.

Formal Definition

In other words: if an element were related to two different elements, they would actually have to be the same. This excludes the 'multi-output' situation.

Examples of Functional Relations

• The function f(x) = x²: each input x corresponds to exactly one output x².
• The 'capital of country' relation: each country has exactly one capital.
• The 'birth year of person' relation: each person has exactly one birth year.

Counterexamples (Non-Functional Relations)

• The 'siblings of person' relation: a person can have multiple siblings, so not functional.
• The 'cities of country' relation: a country has multiple cities, not just one.
• The 'students of class' relation: a class has multiple students.

Connection to Functions

The concept of functional relation is the basis of functions. If a relation is functional, it can be considered a function if, in addition, every starting element has an output (not just at most one), then we speak of a full function.

Summary

The essence of a functional relation is that each input can correspond to at most one output. This is a fundamental property of mathematical functions and plays an important role in everyday life as well, when it comes to unambiguous assignments.

Practice Exercise