Types of Functions in Discrete Mathematics: Injective, Surjective, and Bijective

Explore the fundamental types of functions in discrete mathematics: injective (one-to-one), surjective (onto), and bijective (one-to-one correspondence). This tutorial provides clear definitions, illustrative examples, and explanations to help you understand these key concepts.

Types of Functions in Discrete Mathematics

Functions map input values (from a domain) to output values (in a range or codomain). Different types of functions exist depending on how many inputs map to each output.

1. Injective (One-to-One) Functions

An injective function maps each element in the domain to a unique element in the codomain. No two different inputs produce the same output.

2. Surjective (Onto) Functions

A surjective function maps each element in the codomain to at least one element in the domain. Every possible output value is used.

(An example of a surjective function is provided in the original text and should be included here.)

Note: In a surjective function, the range (the set of actually used output values) equals the codomain.

3. Bijective (One-to-One Correspondence) Functions

A bijective function is both injective (one-to-one) and surjective (onto). Each input maps to a unique output, and every output is used. These functions are invertible (have an inverse function).

(An example of a bijective function is provided in the original text and should be included here.)

4. Into Functions

An into function is a function where not every element in the codomain is mapped to from the domain. The range is a proper subset of the codomain.

(An example of an into function is provided in the original text and should be included here.)

5. One-to-One Into Functions

A one-to-one into function is injective but not surjective (into). Each input maps to a unique output, but not all possible outputs are used.

(An example of a one-to-one into function is provided in the original text and should be included here.)

6. Many-to-One Functions

A many-to-one function is where multiple inputs map to the same output. It's not injective.

(An example of a many-to-one function is provided in the original text and should be included here.)

7. Many-to-One Into Functions

A many-to-one into function is both many-to-one and into (not surjective). Multiple inputs map to the same output, and not all outputs are used.

(An example of a many-to-one into function is provided in the original text and should be included here.)

8. Many-to-One Onto Functions

A many-to-one onto function is both many-to-one and onto (surjective). Multiple inputs can map to the same output, but every possible output value is used.

(An example of a many-to-one onto function is provided in the original text and should be included here.)

Conclusion

These function types are fundamental in discrete mathematics, providing a framework for classifying and analyzing relationships between sets.