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Chapter 7: Problem 15

Show that the set of all bit strings (strings of 0 's and 1's) is countable.

Short Answer

Expert verified
To show that the set of all bit strings, B, is countable, we can systematically list them out and match each bit string to a unique non-negative integer: 1. Group bit strings by length 2. Enumerate bit strings in the sequence For example: 0. "" 1. 0 2. 1 3. 00 4. 01 ... Since every bit string can be found in this list and is matched with a unique non-negative integer, we have a one-to-one correspondence, proving that B is countable.

Step by step solution

01

Define the set of all bit strings

Let B be the set of all bit strings, which includes strings of varying lengths, made up of 0's and 1's.
02

Arrange the strings in sequence

To show that B is countable, we can arrange the strings in a sequence by grouping them according to their length. We will start from the shortest bit strings and move towards longer ones: 1. No bit strings (empty string): "" 2. Length 1: 0, 1 3. Length 2: 00, 01, 10, 11 4. Length 3: 000, 001, 010, 011, 100, 101, 110, 111 5. Length 4: 0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111 We can continue this process indefinitely.
03

Create a one-to-one correspondence with non-negative integers

Now, we can enumerate the sequence to create a one-to-one correspondence with the non-negative integers: 0. "" 1. 0 2. 1 3. 00 4. 01 5. 10 6. 11 7. 000 8. 001 9. 010 10. 011 11. 100 ... 12. 101 ... And so on, with every bit string being assigned a unique non-negative integer.
04

Prove B is countable

As we can list out B systematically and match every element of B to a unique non-negative integer, we have shown that B is countable. This is because a one-to-one correspondence with a countable set, like the non-negative integers, is a key property of countable sets. Therefore, the set of all bit strings is countable.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Bit Strings
A bit string is a sequence of bits, where each bit is a binary digit that can either be 0 or 1. Bit strings are fundamental in computer science because they represent the most basic form of data and can be used to encode information. For example, the bit string '1010' represents a specific sequence of off and on states in a digital system.

In the context of our exercise, we consider all possible bit strings of any length. This includes the empty string, which contains no bits, and strings of increasing length. Understanding the concept of bit strings is essential for further exploration into their countability and their relationship with discrete mathematics.
One-to-One Correspondence
In mathematics, particularly in set theory, a one-to-one correspondence (also called a bijection) is a relationship where each element of one set is paired with exactly one unique element of another set, and vice versa. This concept is crucial when determining the size or cardinality of sets.

For the set of bit strings to be considered countable, we must establish such a correspondence between the set of bit strings and the set of non-negative integers. If we can pair each bit string with a unique non-negative integer without any omissions or repetitions, we will have shown that there is a one-to-one correspondence, which is a vital step in proving that the set of bit strings is countable.
Non-Negative Integers
The term non-negative integers refers to the set of all whole numbers that are greater than or equal to zero. This includes zero itself as well as all the positive integers (1, 2, 3, and so on). Non-negative integers are a subset of the integers and are used extensively in various fields of mathematics, including counting and measuring.

Non-negative integers have an innate order, starting from 0 and increasing by 1 each time. As we assign a unique non-negative integer to each bit string in the exercise, we can confirm that the set is countable as it shows that there is no bit string we cannot count or assign a number to.
Discrete Mathematics
Discrete Mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. This means that in discrete mathematics, we study structures that have distinct and separate values. It includes topics such as logic, set theory, combinatorics, graph theory, and algorithms. Discrete mathematics is fundamental to computer science as it underpins the operations of digital systems.

Our exercise, which explores the countability of the set of all bit strings, falls under the umbrella of discrete mathematics because we deal with distinct entities (bit strings) and use concepts from set theory to reason about them. By showing that the set of all bit strings mirrors the structure of non-negative integers, students get practical insights into the principles of discrete mathematics and its applications in counting and representation in digital systems.

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Most popular questions from this chapter

In each of 16-19 a function \(f\) is defined on a set of real numbers. Determine whether or not \(f\) is one-to-one and justify your answer. $$ f(x)=\frac{x}{x^{2}+1}, \text { for all real numbers } x $$

Draw arrow diagrams for the Boolean functions defined by the following input/output tables. a. \begin{tabular}{|cc|c|} \hline \multicolumn{2}{|c|}{ Input } & Output \\ \hline \(\boldsymbol{P}\) & \(\boldsymbol{Q}\) & \(\boldsymbol{R}\) \\ \hline 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \hline \end{tabular} b. \begin{tabular}{|ccc|c|} \hline \multicolumn{3}{|c|}{ Input } & Output \\ \hline \(\boldsymbol{P}\) & \(\boldsymbol{Q}\) & \(\boldsymbol{R}\) & \(\boldsymbol{S}\) \\ \hline 1 & 1 & 1 & 1 \\ 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \hline \end{tabular}

Show that within any set of thirteen integers chosen from 2 through 40 , there are at least two integers with a common divisor greater than \(1 .\)

Use the definition of logarithm to prove that for any positive real number \(b\) with \(b \neq 1, \log _{\hat{a}} 1=0\).

Explain how it follows from the definition of logarithm that a. \(\log _{b}\left(b^{x}\right)=x\), for all real numbers \(x\). b. \(b^{\log _{b} x}=x\), for all positive real numbers \(x\).
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