Question

There are \(m\) functions from a one-element set to the set \(\\{1,2, \ldots, m\\}\). How many functions are there from a two-element set to \(\\{1,2, \ldots, m\\}\) ? From a three-element set? Give a recurrence for the number \(T(n)\) of functions from an \(n\)-element set to \(\\{1,2, \ldots, m\\}\). Solve the recurrence.

Short answer

The number of functions from an n-element set to \\(\{1, 2, \ldots, m\}\\) is \\(m^n\\).

Step-by-step solution

Calculate Functions from a One-Element Set

For a set with one element \(X = \{x\}\), each function maps the single element to one of the elements in the set \(\{1, 2, \ldots, m\}\). Thus, there are \(m\) possible functions. So for a one-element set, the number of functions is \(m\).

Calculate Functions from a Two-Element Set

When the set has two elements, say \(X = \{x, y\}\), each element in \(X\) can map to any of the \(m\) elements in \(\{1, 2, \ldots, m\}\). The mappings for each element are independent. Therefore, the total number of functions is \(m \times m = m^2\).

Calculate Functions from a Three-Element Set

For a three-element set \(X = \{x, y, z\}\), each of the three elements can independently map to any one of the \(m\) elements in \(\{1, 2, \ldots, m\}\). The number of functions is \(m \times m \times m = m^3\).

Generalize for a Set with n Elements

For an n-element set, each element can be mapped independently to any of the \(m\) elements. Thus, there are \(m^n\) functions from an n-element set to \(\{1, 2, \ldots, m\}\).

Establish a Recurrence Relation

Let \(T(n)\) denote the number of functions from an n-element set to \(\{1, 2, \ldots, m\}\). Then, \(T(n) = m \cdot T(n-1)\) since each new element added to the set multiplies the number of possible mappings by \(m\).

Solve the Recurrence Relation

The recurrence relation is \(T(n) = m \cdot T(n-1)\) with base case \(T(1) = m\). This follows a pattern of \(T(n) = m^n\), confirmed by computing several terms: \(T(1) = m^1, T(2) = m^2, T(3) = m^3\), and so on.

Key concepts

Recurrence Relations

In discrete mathematics, recurrence relations are equations that define sequences of values. These sequences are formulated by applying specific rules that relate each term to its predecessors. For example, the Fibonacci sequence, where each term is the sum of the two preceding ones, is one of the most famous examples of a recurrence relation.

For the exercise at hand, we derived a recurrence relation to calculate the number of functions from a set with a certain number of elements to a target set. The relation is given as \(T(n) = m \cdot T(n-1)\). This tells us that for every additional element in the domain set, the number of possible functions increases multiplicatively by \(m\), the size of the target set.

Recurrence relations offer a way to understand how quantities evolve as their input parameters change. They help in evaluating a sequence without the need to compute every term from scratch. Once we establish the relation, we can predict further terms in the sequence by simply using previous results.

Sets and Functions

Sets and functions are fundamental in discrete mathematics, playing a crucial role in various applications. A set is a collection of distinct objects, which can be anything from numbers to symbols or even other sets.

Functions are mappings from one set (called the domain) to another set (called the codomain). Each element from the domain must be associated with exactly one element in the codomain. In the given exercise, we deal with sets containing up to \(n\) elements and a codomain set \(\{1, 2, \ldots, m \}\).

The concept of sets and functions is powerful since it allows us to model real-world situations and discrete objects effectively. For instance, we can use functions to describe how data points move to new data points across various transformations. Understanding how to structure domains and codomains is key to recognizing how functional mapping operates.

Function Mapping

Function mapping is the process by which each element of a set (domain) corresponds to an element in another set (codomain). The exercise examines how many such mappings exist between a small set (like one with \(n\) elements) and a larger codomain set \(\{1, 2, \ldots, m\}\).

For any function \(f: A \rightarrow B\), each element in the set \(A\) can map to any element in set \(B\). Thus, to determine the total number of functions possible, we raise the number of possible codomain mappings \(m\) to the power of the number of elements in the domain set \(n\). This results in \(m^n\) total possible mappings.

Function mapping helps clarify the relationships between different sets and aids in visualization of more abstract concepts such as transformations and permutations. By understanding function mappings, students can grasp more complex structures and operations in mathematics, enhancing their ability to solve problems that involve sets and functions.