Injective Relation (Injective Function)

A relation or function is called injective (or one-to-one) if different inputs always correspond to different outputs. This means that it never happens that two different inputs are connected to the same output.

Formal Definition

In other words: if two elements were connected to the same output, they would actually be identical. This ensures that outputs do not 'repeat' for different inputs.

Examples of Injective Relations

• The assignment f(x) = 2x on real numbers: different x values always correspond to different 2x values.
• People's personal ID numbers: every person has a unique identifier.
• Cars' license plates: every car has a unique license plate.

Counterexamples (Non-Injective Relations)

• The assignment f(x) = x² on integers: f(2) = 4 and f(-2) = 4, so two different inputs correspond to the same output.
• People's hair color: multiple people can have the same hair color.
• Countries' official languages: multiple countries can use the same language.

Relation to Functions

Injectivity is one of the possible properties of functions. A function can be injective, surjective, or bijective. Injectivity guarantees the 'uniqueness' direction: every different input goes to a different output.

Summary

The essence of an injective relation or function is that there are no two different inputs that lead to the same output. This one-to-one correspondence is crucial in many mathematical and computer science fields.

Practice Exercise