Problem 14 - Discrete Mathematics with Applications

Question

Let \(\Sigma=\\{0,1\\} .\) Let \(f: \Sigma^{*} \rightarrow \mathbf{W}\) defined by \(f(x)=\|x\| .\) Evaluate \(f(x)\) for each value of \(x\). $$00110011$$

Step-by-Step Solution

Answer

For the given bit string x = 00110011, we need to evaluate the function f(x) by finding the length of x. Since there are 8 bits in the given bit string, f(x) = 8.

Step 1: Write down the given bit string x

We are given x = 00110011 as the input bit string.

Step 2: Count the number of bits in x

To evaluate f(x), we need to count the number of bits in the given bit string x. We can easily see that x has 8 bits.

Step 3: Evaluate the function f(x)

We know that f(x) is the length of the given bit string x. As we have counted 8 bits in the given bit string, f(x) = 8. In conclusion, after evaluating the function f for the given input bit string x = 00110011, we find that f(x) = 8.

Key Concepts

Understanding Bit Count

Bit count refers to the number of individual binary digits, or 'bits', present in a string composed of 0s and 1s. In the context of binary strings like the example given, each digit is a bit. For example, in the string 00110011, we work with:

• Each '0' and '1' counts as a single bit.
• The entire string is composed of 8 bits.

Counting bits is a fundamental aspect of understanding binary data, as it serves as the basic unit of information in computing. To count bits, simply iterate through each character of the string and tally them up. This concept helps us measure or quantify the length and information capacity of any given binary sequence.

Exploring String Length

The length of a string in computing refers to the total count of characters or elements it contains. With binary strings made up of just '0's and '1's, the string length is equal to its bit count.
In our specific example, the string 00110011 is composed of 8 characters, meaning its string length is 8. This notion of string length is significant because:

• It provides insight into the data size being handled.
• It is a direct representation of memory usage in binary systems.

Knowing the string length allows us to effectively use functions to manipulate or interpret binary sequences, giving us more control over computational processes.

Function Evaluation Explained

Function evaluation in mathematics and computer science is the process of calculating the output of a function given an input. Here, the function f is defined as producing the length of a given binary string, denoted by \(f(x) = \|x\|\).
For the binary string 00110011, we evaluate this function by determining its length. Since we already counted the 8 bits involved, the evaluation results in:

• f(x) = 8

This not only lets us find the length of the string but also understand how functions can map inputs to outputs based on defined rules. It helps when interpreting various data types and structures, adjusting them to meet computational demands. Function evaluation can be applied in numerous contexts beyond simple string length calculation, such as encryption, data formatting, and error detection.