Question

Let \(X=\\{a, b\\}\) (a) There are nine partial functions \(F: X \rightarrow X\). List them. (b) There are four functions \(F: X \rightarrow X\). List them. (c) List all \(l-l\) functions \(F: X \rightarrow X\). (d) List all onto functions \(F: X \rightarrow X\).

Short answer

(a) 9 partial functions, (b) 4 total functions, (c) 2 1-1 functions, (d) 2 onto functions.

Step-by-step solution

Understanding Partial Functions

A partial function from set \( X \) to set \( X \) means that every element in the domain (also \( X \)) may or may not have an image in the codomain (which is also \( X \)). Given \( |X| = 2 \), any element in \( X \) has three choices: map to 'a', map to 'b', or map to nothing. Therefore, there are \( 3 \times 3 = 9 \) partial functions.

List of Partial Functions

The partial functions from \( X \) to \( X \) can be listed as: 1. \( \{ (a, a), (b, a) \} \) 2. \( \{ (a, a), (b, b) \} \) 3. \( \{ (a, b), (b, a) \} \) 4. \{(a, b), (b, b) \} \ 5. \{(a, a) \}\ 6. \{(a, b) \}\ 7. \{(b, a) \}\ 8. \{(b, b) \}\ 9. \{ \} \.

Understanding Total Functions

A total function from \( X \) to \( X \) must map every element of \( X \) to some element in \( X \). Thus, each of the two elements in \( X \) ('a' and 'b') can independently map to either 'a' or 'b', leading to \( 2 \times 2 = 4 \) functions.

List of Total Functions

Functions that are total (and therefore everywhere defined) from \( X \) to \( X \) are: 1. \( \{ (a, a), (b, a) \} \) 2. \{(a, a), (b, b) \}\ 3. \{(a, b), (b, a) \}\ 4. \{(a, b), (b, b) \}.

Understanding 1-1 Functions

A one-to-one (injective) function means no two elements in the domain map to the same element in the codomain. For \( F: X \rightarrow X \), this requires \( |X| = |X| \), hence either identity or a switch is valid.

List of 1-1 Functions

For the set \( X \), only the two permutations are: 1. \( \{(a, a), (b, b)\} \) - identity function, 2. \( \{(a, b), (b, a)\} \) - swap mappings.

Understanding Onto Functions

A surjective or onto function covers the codomain entirely. Every element of \( X \) as a codomain must be mapped by some element from the domain \( X \). This results in similar constraints to 1-1 functions for sets of the same size.

Listing Onto Functions

Matching the domain and codomain: 1. \( \{(a, a), (b, b)\} \) - each maps to itself, and 2. \( \{(a, b), (b, a)\} \) - each maps to the other, meeting the surjectivity requirement.

Key concepts

Partial Functions

A partial function is a specific type of function that may not have a defined output for every input in its domain. In the context of discrete mathematics, understanding partial functions involves realizing that each input might map to one of the elements in the codomain or not map to anything at all.
For example, consider a set \(X = \{a, b\}\). A partial function \(F: X \rightarrow X\) allows each element in \(X\) to either:

• Map to 'a'
• Map to 'b'
• Not map at all

This gives us the flexibility of mapping some or no elements entirely. For a set of size 2, the number of possible partial functions is \(3^2 = 9\).
These partial functions highlight the concept that not every function must be 'everywhere defined', giving room for undefined mappings, which is crucial when dealing with partial data sets or incomplete relationships.

Total Functions

In contrast to partial functions, total functions require that every element in the domain has an output in the codomain. This means a total function \(F: X \rightarrow X\) must map every element of set \(X\) to a specific element in the codomain, without exception.
For a set \(X = \{a, b\}\), this obligation creates a firm rule: each element ('a' and 'b') in the domain must map into \(X\), resulting in a clear mapping from each domain element to some codomain element. With each having two available targets, we find there are \(2 \times 2 = 4\) unique total functions.
These total functions demonstrate a fundamental principle in mathematics where each input is accounted for, ensuring that no element is 'left behind'. The guaranteed mappings are used in systems requiring complete and non-optional associations between datasets.

Injective Functions

Injective functions, also known as one-to-one functions, are those in which no two different elements from the domain map to the same element in the codomain. This ensures every element in the codomain is the image of at most one element from the domain.
When set \(X = \{a, b\}\), the requirement for an injective function means that each element from the domain 'a' and 'b' maps uniquely to different elements in the codomain. Thus, the possible injective functions include:

• The identity function \(\{(a, a), (b, b)\}\)
• The swap function \(\{(a, b), (b, a)\}\)

These mappings preserve uniqueness across sets of equal size, which is valuable in scenarios where distinct identification is crucial, such as in data management and cryptography.

Surjective Functions

Surjective functions, also known as onto functions, ensure that every element of the codomain is mapped by at least one element from the domain. This is different from injective functions, which emphasize uniqueness of mappings rather than coverage.
For a set \(X = \{a, b\}\), being surjective means ensuring the codomain is fully "covered" by mappings from the domain. Our surjective functions for this small set are:

• \(\{(a, a), (b, b)\}\) - Identity mapping
• \(\{(a, b), (b, a)\}\) - Completeness through swapping

Such functions are crucial in ensuring that relationships or transformations account for all possible outputs, often used in fields like computer science to design algorithms where every potential outcome must be ensured.