Surjective Relation (Mapping)

We call a relation or function surjective (or onto, in English: surjective) if every element in the target set has at least one input connected to it. In other words: there is no element in the target set that the function does not 'reach'.

Formal Definition

This expresses that for every element in the target set, we find at least one starting element that the function maps to it.

Examples of Surjective Relations

• The f(x) = x³ on real numbers: every real number is the cube of some real number.
• The 'birth month of people' relation: every month has at least one person born in it.
• The identity function f(x) = x: every element in the target set is reached exactly once.

Counterexamples (Non-Surjective Relations)

• The f(x) = x² on real numbers: negative numbers are not reached (no real square root).
• The 'capitals of countries to cities' relation: many cities are not capitals.
• The f(x) = 2x on integers: odd numbers are not reached.

Connection to Functions

Surjectivity ensures that the function covers every element in the target set. A function can be surjective, injective, or bijective. Surjectivity emphasizes 'completeness': no target element is left uncovered.

Summary

The essence of a surjective relation is that every target set element corresponds to at least one input. This is important in mathematics and computer science, for example, when ensuring that every possible value can be produced by a function.

Practice Exercise