Problem 7 - Applied Discrete Structures

Question

Let \(f, g,\) and \(h\) all be functions from \(\mathbb{Z}\) into \(\mathbb{Z}\) defined by \(f(n)=n+5\), \(g(n)=n-2,\) and \(h(n)=n^{2} .\) Define: (a) \(f \circ g\) (b) \(f^{3}\) (c) \(f \circ h\)

Step-by-Step Solution

Answer

(a) \( f \circ g = n + 3 \) (b) \( f^3 = n + 15 \) (c) \( f \circ h = n^2 + 5 \)

Step 1: Understanding Function Composition

Function composition involves applying one function to the result of another. For example, the composition \( f \circ g \) means apply \( g \) first and then apply \( f \) to the result.

Step 2: Calculate \( f \circ g \)

Start by applying function \( g \) to \( n \). We have \( g(n) = n - 2 \). Now, apply function \( f \) to \( g(n) \):\[(f \circ g)(n) = f(g(n)) = f(n - 2) = (n - 2) + 5 = n + 3.\]So, \( f \circ g = n + 3 \).

Step 3: Understanding Function Iteration

Function iteration means applying the same function multiple times. Here \( f^3 \) means applying \( f \) three times sequentially, i.e., \( f(f(f(n))) \).

Step 4: Calculate \( f^3 \)

Start by applying \( f \) to \( n \):\( f(n) = n + 5 \).Next, apply \( f \) again to the result:\[ f(f(n)) = f(n + 5) = (n + 5) + 5 = n + 10. \]Finally, apply \( f \) once more:\[ f(f(f(n))) = f(n + 10) = (n + 10) + 5 = n + 15. \]Thus, \( f^3(n) = n + 15 \).

Step 5: Calculate \( f \circ h \)

Apply function \( h \) to \( n \), which gives \( h(n) = n^2 \). Now apply function \( f \) to \( h(n) \):\[(f \circ h)(n) = f(h(n)) = f(n^2) = n^2 + 5.\]So, \( f \circ h = n^2 + 5 \).

Key Concepts

Function Iteration

Function iteration is a process of applying a single function multiple times in sequence. Consider for instance the function \( f \) with the rule \( f(n) = n+5 \). When we talk about \( f^3 \), it refers to applying function \( f \) three times to an initial input. The sequence is as follows:

• Start with applying \( f \) to \( n \), giving us \( f(n) = n + 5 \).
• Next, apply \( f \) again to the result, producing \( f(f(n)) = (n+5) + 5 = n + 10 \).
• Finally, apply \( f \) once more to this result, yielding \( f(f(f(n))) = (n+10) + 5 = n + 15 \).

With function iteration, recognizing patterns can be useful. For \( f^k \), where \( k \) is the number of iterations, the function essentially adds \( 5k \) to the initial integer. This method particularly helps simplify complex problems in both algebra and programming where looping through function operations is necessary.
Understanding function iteration is vital, especially for algorithms that require repetitive operations or in mathematics, where finding a general formula for repeated applications can simplify otherwise complex solutions.

Discrete Mathematics

Discrete mathematics plays a significant role when dealing with functions like the ones in the exercise, especially because it typically involves countable, distinct elements. Unlike continuous mathematics, which deals with smoothly varying quantities, discrete mathematics focuses on integers, graphs, logical statements, and other countable structures.
In terms of functions in discrete mathematics, think about how operations are applied over set elements. For instance, function composition and iteration are examples of how elements are systematically manipulated, following definite rules.
In our example, when you compose \( f \) and \( g \), or iterate \( f \), you essentially manipulate integers in a stepwise manner using arithmetic operations.

• Composition: In \( f \circ g \), apply one function and then the other, leading to a new function \( n + 3 \).
• Iteration: With \( f^3 \), apply the function repeatedly, resulting in \( n + 15 \).

Discrete mathematics helps analyze these processes systematically, whether you're working on algorithms, mathematical proofs, or solving logical problems. It provides the tools and structures necessary to understand how these functions behave over the integers, ensuring precision and consistency in results.

Integer Functions

Integer functions are functions that map integers to integers, just like the ones defined in the exercise with \( f(n) = n + 5 \), \( g(n) = n - 2 \), and \( h(n) = n^2 \). These functions are crucial in numerous mathematical areas including number theory, cryptography, and computer science.

• Adding integers, like in \( f(n) \), changes the original value by a constant amount.
• Subtractive functions, like \( g(n) \), reduce the integer's value.
• Quadratic functions, such as \( h(n) = n^2 \), map integers to their squares, representing non-linear growth.

When working with integer functions, each function’s specific rule is applied to any given integer input. This methodology ensures each integer always maps to another integer, maintaining the integrity of the discrete system.
The predictable nature of integer functions is particularly useful. For example, when composing functions, such as \( f \circ h = n^2 + 5 \), you combine operations like addition and squaring into seamless execution, transforming the integer inputs systematically. Through their structured methods, integer functions support thorough analysis or computations, which are foundational in both theoretical and practical applications.