Problem 12 - Discrete Mathematics with Applications

Question

Let \(\Sigma=\\{0,1\\}\) . Let \(f : \Sigma^{*} \rightarrow\) W defined by \(f(x)=\|x\| .\) Evaluate \(f(x)\) for each value of \(x .\) 1010100

Step-by-Step Solution

Answer

The given function is \(f(x)=\|x\|\), which calculates the length of the input string x. The provided input string x is 1010100. The length of the input string x is 7. Therefore, \(f(x) = \|x\| = 7\).

Step 1: Understand the function definition

The given function \(f(x)=\|x\|\) calculates the length of the input string x, where x is composed of 0's and 1's.

Step 2: Identify the input string

For this exercise, the provided input string x is 1010100.

Step 3: Calculate the length of the input string

The given input string x contains 7 symbols (1, 0, 1, 0, 1, 0, and 0). Thus, the length of the input string x is 7.

Step 4: Evaluate the function using the calculated length

Now that we have determined the length of the input string x, we can evaluate the function \(f(x)\) by substituting the length as follows: \(f(x) = \|x\| = 7\). The result of \(f(x)\) for the given input string is 7.

Key Concepts

Function Evaluation

A function can be seen as a mathematical entity where each input is associated with exactly one output. Function evaluation is the process of finding this output for a given input by applying the function's rule. In other words, we apply the function to a specific input to see what output it gives.

In our particular scenario, we are dealing with the function \( f : \Sigma^{*} \rightarrow W \) defined as \( f(x) = \|x\| \). This means that when we evaluate \( f \) for any string \( x \), the function will return the length of that string.

• The input \( x \) is a sequence of binary digits (0s and 1s).
• The output \( f(x) \) is simply the count of characters in \( x \).

This method of evaluation is quite straightforward, as it involves counting the elements of the string to generate what is typically a non-negative integer output.
Understanding function evaluation can save you time and clarify how these mappings (or functions) work in discrete mathematics.

Binary Strings

Binary strings are sequences composed entirely of the digits 0 and 1. They lie at the heart of computer science and digital communications because computers fundamentally operate in binary. But they are also crucial in discrete mathematics for representing sets, functions, and more.

In the problem solved here, the binary string is '1010100'. Such strings can represent anything from data or code to numbers, depending on the context.

• Binary strings can be of any length: Some might be as short as a single digit, while others might extend to thousands of digits.
• Operations on binary strings include concatenation, evaluating specific functions, or manipulating individual bits.

These versatile sequences provide a simple yet powerful way to handle data and logical operations, such as within our function \( f(x)=\|x\| \), which operates on binary strings by evaluating their length.

String Length

String length is a measure of the number of characters in a string, which in this case refers to the total count of binary digits within a binary string. It can indicate how much information or data a string holds and is a straightforward yet vital concept in computer science and mathematics.

For the binary string '1010100', the length function \( \|x\| \) reveals its length by counting each binary digit.

• Counting the '1's and '0's gives a total of 7, which is the length of the string.
• In certain contexts, knowing the string length can be critical for efficient storage, processing, and transmission of binary data.

This concept allows us to quantify elements of a string and is especially useful in the evaluation of functions like \( f(x) \), which in this problem, naturally gives us 7 as the length of '1010100', following the simple process of enumerating its characters.