Problem 32 - 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 \(w\). nesthesi

Step-by-Step Solution

Answer

Given the word \(w\) = "nesthesi", we can calculate the function \(g(w) = awa\) as follows: Step 1: Prefix "a" to the word \(w\): "anesthesi" Step 2: Suffix "a" to the result of Step 1: "anesthesia" Hence, \(f(\text{"nesthesi"}) = \text{"anesthesia"}\).

Step 1: Consider the given word \(w\)

Given a string \(w\), first, make sure it is from the English alphabet. Otherwise, the function \(g\) cannot be applied.

Step 2: Prefix the word with the letter "a"

Now, it's time to apply the first part of the function: adding the prefix "a" to the word \(w\). Simply add the letter "a" to the beginning of the word. Example: If \(w\) is "nesthesi", the result is "anesthesi".

Step 3: Suffix the word with the letter "a"

Now, apply the second part of the function: adding the suffix "a" to the word obtained in the previous step. Simply add the letter "a" at the end of the word. Example: If the word is "anesthesi" from the previous step, the result is "anesthesia".

Step 4: Obtain the result, \(f(w)\)

Now that both the prefix and suffix "a" have been added to the given word \(w\), we have the final result, \(f(w)\). Example: \(f(\text{"nesthesi"}) = \text{"anesthesia"}\). Apply the steps above to any given word \(w\) to find \(f(w)\).

Key Concepts

Functions and Mappings

In discrete mathematics, a function is a relation between a set of inputs and a set of possible outputs with the property that each input is related to exactly one output. This is commonly expressed as f: X \rightarrow Y, denoting that function f maps elements from set X to set Y. When dealing with mappings, it involves the process of assigning an element from the domain (input set) to an element in the codomain (output set).

For example, in the given exercise, we have a function g which maps strings from the domain \(\Sigma^{*}\), the set of all strings over the English alphabet, to the codomain which is also \(\Sigma^{*}\). The function is well-defined as for every word \( w \) from the domain, there is a unique word \( f(w) \) in the codomain produced by prefixing and suffixing the word with the letter 'a'.

This function can be visualized as taking each word, symbolically represented as a point in the domain, and 'mapping' it to a new word in the codomain after the specified operation is performed.

String Manipulation

In the context of our function, string manipulation involves altering or operating on a sequence of characters. This is a foundational concept in programming and computer science, but also slips into various mathematical functions dealing with strings.

The operation defined by our function \( f(w) = \) 'awa' involves two basic string manipulation actions: prefixing and suffixing. To prefix a string means to add another string at the beginning, while to suffix means to add it at the end. The outcome is a new string that includes the original string precisely encapsulated with the character 'a'.

With our given word 'nesthesi', the process of string manipulation is simple:

• Prefix 'nesthesi' with 'a' to get 'anesthesi'.
• Suffix 'anesthesi' with 'a' to get 'anesthesia'.

Students should practice these operations to ensure a complete understanding of how string manipulation works within the scope of mathematical functions.

Function Composition

When we talk about function composition, we refer to the combination of two functions in such a way that the output of one function becomes the input of the other. It is denoted as \( (f \: \: o \: \: g)(x) \) or simply \( f(g(x)) \), meaning you first apply function \( g \) to \( x \) and then apply function \( f \) to the result of \( g(x) \).

In the context of the given exercise, while function composition isn't explicitly used, it's implied in the operation where 'awa' is created by first applying the prefix 'a' and then the suffix 'a'. If we consider these as separate functions, \( p(w) \) for prefix and \( s(w) \) for suffix, we can express the original function \( f(w) \) as a composition: \( f(w) = s(p(w)) \).

Understanding function composition is vital for more complex mathematical operations and is an essential concept for both mathematics and computer science. By mastering function composition, students can solve more intricate problems by breaking them down into simpler, sequential steps.