## Definition of Discrete Functions

A **discrete function** is a function that is defined only for a set of discrete points. Unlike continuous functions that have outputs for every input within a certain range, discrete functions are defined at specific and separate values.

### Characteristics of Discrete Functions

Discrete functions have several key characteristics:

- Defined only at specific, isolated points.
- The graph consists of isolated points, not a continuous line.
- Examples include functions defined on integers or specific subsets of real numbers.

### Examples of Discrete Functions

Consider the function that maps each integer to its square:

\[ f(x) = x^2 \]

This function is defined only for integer values of *x*. Therefore, *f* is a discrete function.

**Example:**

If you take the input values *x = -2, -1, 0, 1, 2*, the corresponding output values are:

x | f(x) |

-2 | 4 |

-1 | 1 |

0 | 0 |

1 | 1 |

2 | 4 |

A function defined only for distinct and separate points is a **discrete function**.

Remember, not all functions defined on integers are discrete. The key is that the function is only defined at isolated points.

When dealing with discrete functions, you might encounter sequences. A **sequence** is a type of function whose domain is the set of natural numbers. Sequences can be finite or infinite. One of the most famous is the Fibonacci sequence defined as:

\[ F(n) = F(n-1) + F(n-2) \]

with the initial conditions:

\[ F(0) = 0, \] \[ F(1) = 1 \]

This sequence is defined only for non-negative integers, making it a discrete function.

## Functions in Discrete Mathematics

In discrete mathematics, functions are a fundamental concept used to map elements from one set to another. Discrete functions are a special kind of function where the input values are distinct and separate points.

### Characteristics of Discrete Functions

There are some unique characteristics that differentiate discrete functions from continuous functions:

**Isolation:**The function is defined only at distinct, separate points.**Graph:**The graph of a discrete function consists of isolated points rather than a continuous line.**Examples:**Functions defined on integers or specific subsets of real numbers.

### Examples of Discrete Functions

To better understand discrete functions, let's consider some examples:

**Example:**

Consider the function that maps each integer to its square:

\[f(x) = x^2\]

This function is defined only for integer values of *x*. For instance:

x | f(x) |

-2 | 4 |

-1 | 1 |

0 | 0 |

1 | 1 |

2 | 4 |

A deeper look into sequences is essential when discussing discrete functions. A **sequence** is a function where the domain is the set of natural numbers. This means that each natural number maps to a specific element in the sequence.

One of the most famous sequences is the Fibonacci sequence, defined as:

\[F(n) = F(n-1) + F(n-2)\]

with initial conditions:

\[F(0) = 0\]

\[F(1) = 1\]

The Fibonacci sequence is defined only for non-negative integers, making it a discrete function.

Remember, not all functions defined on integers are discrete. Isolated definition is the key.

A function defined only for distinct and separate points is a **discrete function**.

### Real-world Applications of Discrete Functions

Discrete functions have numerous applications in real-world scenarios:

**Computer Science:**Algorithms often use discrete functions to process data structured in lists or arrays.**Cryptography:**Functions defined on discrete points are crucial for encryption and decryption processes.**Inventory Management:**Businesses use discrete functions to manage stock levels, tracking items in integer quantities.

**Example:**

In computer science, a hash function is a discrete function used for efficiently finding data. The function maps input data (keys) to specific locations in a hash table:

\[h(k) = k \, \text{mod} \, N\]

where *k* is the key, and *N* is the number of slots in the hash table.

Discrete functions are foundational in graph theory, where edges and vertices are analysed using such functions.

## Examples of Discrete Functions in Mathematics

In mathematics, discrete functions play a critical role, especially in areas like computer science, cryptography, and operations research. These functions are defined at distinct, isolated points.

### Understanding Sequences as Discrete Functions

A sequence is a type of discrete function where the domain is the set of natural numbers. For example, the arithmetic sequence can be defined by:

\[a_n = a + (n-1)d\]

where *a* is the first term, *d* is the common difference, and *n* is the position of the term in the sequence.

**Example:**

Consider the arithmetic sequence \(a_n = 2 + (n-1)3\). Here, \(a = 2\) and \(d = 3\).

n | a_{n} |

1 | 2 |

2 | 5 |

3 | 8 |

4 | 11 |

5 | 14 |

### The Use of Discrete Functions in Graph Theory

In graph theory, discrete functions are used to define the relationships between vertices and edges in a graph. Consider a graph representing a network:

**Vertices (Nodes):**Represent the individual entities.**Edges (Links):**Represent the connections between entities.

**Example:**

A simple undirected graph can be defined by the adjacency function:

\[f(u, v) = \begin{cases}1, & \text{if } u \text{ and } v \text{ are connected}\0, & \text{otherwise} \end{cases}\]

Where *u* and *v* are vertices in the graph.

In computer science, the concept of a hash function is crucial. A hash function is a discrete function used to efficiently map data to specific locations in a hash table. An example of a simple hash function could be:

\[h(k) = k \% N\]

where *k* is the key and *N* is the number of slots in the hash table. This function ensures data is stored in an organised manner for quick access.

### Discrete Probability Distributions

Discrete functions also play a significant role in probability theory. A discrete probability distribution describes the probabilities of the outcomes of a discrete random variable. For example, the probability mass function (PMF) of a discrete random variable *X* is:

\[P(X = x)\]

where *x* is a possible value of *X*.

**Example:**

For a fair six-sided die, the PMF is:

\[P(X = x) = \frac{1}{6}, \text{ for } x = 1, 2, 3, 4, 5, 6\]

Remember, in a discrete probability distribution, all probabilities must sum to one.

### Real-world Applications of Discrete Functions

Discrete functions have numerous real-world applications:

**Computer Science:**Algorithms often use discrete functions to process data structured in lists or arrays.**Cryptography:**Functions defined on discrete points are crucial for encryption and decryption processes.**Economics:**Discrete functions are used in modelling economic decisions based on discrete time intervals.

**Example:** In computer science, a hash function is used for efficiently finding data. The function maps input data (keys) to specific locations in a hash table:

\[h(k) = k \% N\]

where *k* is the key, and *N* is the number of slots in the hash table.

Discrete functions are foundational in graph theory, where edges and vertices are analysed using such functions.

## Discrete Functions Exercises

Practising with discrete functions helps you grasp essential mathematical concepts and improves problem-solving skills. Engaging with exercises can consolidate your understanding of how these functions work and their applications.

### Discrete Function Example

Consider an example where a discrete function maps each integer to its factorial value:

\[ f(x) = x! \]

This function is only defined for non-negative integer values of *x*.

**Example:**

For input values \( x = 0, 1, 2, 3, 4 \), the corresponding output values are:

x | f(x) |
---|---|

0 | 1 |

1 | 1 |

2 | 2 |

3 | 6 |

4 | 24 |

Factorial of a number \( n \) is the product of all positive integers up to \( n \): \( n! = n \cdot (n-1) \cdot (n-2) \cdot \, \ldots \cdot 1 \).

In some examples, you can encounter the **recurrence relation**. For instance, the factorial function can be recursively defined as:

\[ f(x) = x \cdot f(x-1) \]

with the base case \( f(0) = 1 \).

### Discrete Function Graph

Graphs of discrete functions help visualise the relationship between the input and output values. Unlike continuous functions, the graph of a discrete function consists of isolated points.

**Example:**

Consider the discrete function \( f(x) = x^2 \), only defined for integer values of *x*. The graph will show a series of isolated points where each point corresponds to a square of an integer.

To plot a discrete function, use a scatter plot, where each point represents a value of the function at a specific input.

Graphs of discrete functions can also reveal properties such as periodicity and symmetry. For example, the function \( f(x) = (-1)^x \) alternates between 1 and -1 for integer values of *x*, showing a clear periodic pattern with a period of 2.

## Discrete functions - Key takeaways

**Definition of Discrete Functions:**A function defined only for distinct, separate points.**Characteristics:**Defined at isolated points; graph consists of isolated points, not a continuous line.**Example:**The function mapping each integer to its square (\f(x) = x^2\for integer values).**Applications:**Used in computer science algorithms, cryptography, inventory management, and graph theory.**Discrete Function Graph:**A graph consisting of isolated points rather than a continuous line, can use a scatter plot to visualise.

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