Problem 30 - Discrete Mathematics with Applications

Question

Let \(\Sigma\) denote the English alphabet. Let \(g: \Sigma^{*} \rightarrow \Sigma^{*}\) defined by \(f(w)=\) awa. The function prefixes and suffixes each word with a. Find \(f(w)\) for each word \(w\). mbrosi

Step-by-Step Solution

Answer

The function \(g(w)=\text{awa}\) takes a word \(w\) and adds an "a" to the beginning and end of the word. Applying this function to the word "mbrosi", we get \(g(\text{mbrosi})=\text{a(mbrosi)a}\), which results in the word "ambrosia".

Step 1: Understand the function g(w)

To find the image of a given word under the function \(g\), it is just needed to add an "a" to the beginning and another "a" to the end of the word. Suppose \(w =\) mbrosi. We can apply function \(g\) to perform the operation as follows.

Step 2: Apply the function g(w) to the word "mbrosi"

In this step, we will take the word "mbrosi" and add an "a" to the beginning and another "a" to the end, according to the definition of function \(g\). \(g(\text{mbrosi}) = \text{a(mbrosi)a}\) By following this step, we find that the image of the word "mbrosi" under the function \(g\) is "ambrosia".

Key Concepts

Functions

In mathematics and computer science, a function represents a specific kind of relationship between two sets. In this context, a function takes an input from one set (the domain) and produces an output in another set (the codomain). A function is essentially a rule that assigns each input exactly one output.

When we talk about "functions" in programming, they operate in a similar manner. They take an input, process it using predefined instructions, and return an output. This process is much like a machine: you input something, it does some work, and outputs something else.

• Functions are used to perform operations consistently, without having to rewrite code.
• They promote code reusability and better organization.
• Understanding the input-output relationship is key to working with functions effectively.

In our example, the function is defined as \( g(w) = a(w)a \), which simply means taking any word \( w \) and appending an 'a' at the beginning and end of the word, thus transforming it functionally.Here, \( g \) is the function and \( w \) is the placeholder for any input word from the set of all possible words composed of letters from the English alphabet.

String Manipulation

String manipulation is a common operation in many programming languages, involving tasks that modify, process, or analyze text-based data. Strings are sequences of characters, and the manipulation of these sequences allows us to format and transform data to meet specific requirements.

• Common operations include adding, cutting, or replacing parts of a string.
• String manipulation is highly useful in preparing data for display or storage.

In our given exercise, the task was to "manipulate" the string by adding specific characters at defined positions. The function \( g(w) = a(w)a \) illustrates a basic form of string manipulation by concatenating 'a' to both the beginning and end of the input string "mbrosi."

This simple operation can be extended to more complex manipulations including reversing strings, changing case, or extracting substrings. Efficient string manipulation is crucial in optimizing the performance of software applications because strings are among the most used data types.

English Alphabet

The English alphabet consists of 26 letters ranging from A to Z. It serves as the foundation for English text and can be treated as a set in mathematical and computational contexts, often denoted as \( \Sigma \).

• In computer science, letters of the alphabet are used to form strings (sequences of characters).
• Alphabetic systems are used in defining string operations and functions, as seen in our exercise.

When dealing with functions like \( g(w) \) that manipulate strings, the alphabet limits the set of characters that can be used. Thus, our input "mbrosi" is a word that conforms to the use of valid characters from the English alphabet. By prefixing and suffixing the word with an 'a', we create a new word "ambrosia," which still belongs to the set of strings that can be constructed using the English alphabet.

Understanding alphabetic constraints helps in designing algorithms and functions that operate seamlessly across text-based datasets.