Problem 50
Question
Arrange the binary words of the given length in increasing order of magnitude. Length three.
Step-by-Step Solution
Verified Answer
The binary words of length three arranged in increasing order of magnitude are: 000, 001, 010, 011, 100, 101, 110, and 111.
1Step 1: List all binary words of length three
Write down all possible combinations of 0s and 1s for a three-digit binary word. There are 2^3 = 8 possible combinations:
1. 000
2. 001
3. 010
4. 011
5. 100
6. 101
7. 110
8. 111
2Step 2: Arrange the binary words in increasing order
Binary words can be treated as base-2 numbers, which can be compared like ordinary decimal numbers. Convert each binary word to its decimal equivalent and sort them in ascending order:
1. 000 = \(0_{10}\)
2. 001 = \(1_{10}\)
3. 010 = \(2_{10}\)
4. 011 = \(3_{10}\)
5. 100 = \(4_{10}\)
6. 101 = \(5_{10}\)
7. 110 = \(6_{10}\)
8. 111 = \(7_{10}\)
Now that the binary words are arranged in increasing order of their decimal equivalents, they are in the desired order:
1. 000
2. 001
3. 010
4. 011
5. 100
6. 101
7. 110
8. 111
Key Concepts
Discrete MathematicsBinary Number SystemCombinatorics
Discrete Mathematics
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. This means that, unlike calculus which deals with continuous functions and changes, discrete mathematics studies objects that can only take on distinct, separated values.
When dealing with binary words, for instance, discrete mathematics considers a limited set of values: the digits 0 and 1. These binary digits, or bits, can be combined in various discrete ways to form binary words used in computer science and digital technologies. This area of study encompasses a range of topics, including logic, set theory, graph theory, and combinatorics—the latter being a vital part of structuring and counting the distinct configurations of binary words.
The exercise of arranging binary words in increasing order of magnitude involves an understanding of set-defined operations and counting principles, both of which are foundational in discrete mathematics.
When dealing with binary words, for instance, discrete mathematics considers a limited set of values: the digits 0 and 1. These binary digits, or bits, can be combined in various discrete ways to form binary words used in computer science and digital technologies. This area of study encompasses a range of topics, including logic, set theory, graph theory, and combinatorics—the latter being a vital part of structuring and counting the distinct configurations of binary words.
The exercise of arranging binary words in increasing order of magnitude involves an understanding of set-defined operations and counting principles, both of which are foundational in discrete mathematics.
Binary Number System
The binary number system, also known as base-2, is a positional numeral system that uses only two digits: 0 and 1. Each digit in a binary number is known as a bit, and each binary number is a combination of these bits. This system is intrinsic to the digital world and is the basis for computer and other electronic systems' data representation and processing.
In the context of the exercise presented, a three-bit binary number can take on any value from 000 to 111, which in decimal system are ranged from 0 to 7, respectively. Understanding how to convert between binary and decimal numbers is crucial in many aspects of computer science and electronic engineering. This conversion is often done by multiplying each bit by 2 raised to the power of its position from the right and summing these products up.
In the context of the exercise presented, a three-bit binary number can take on any value from 000 to 111, which in decimal system are ranged from 0 to 7, respectively. Understanding how to convert between binary and decimal numbers is crucial in many aspects of computer science and electronic engineering. This conversion is often done by multiplying each bit by 2 raised to the power of its position from the right and summing these products up.
Combinatorics
Combinatorics is the branch of mathematics dealing with combinations, permutations, and the countable structures of sets. It helps determine the number of different ways to arrange or choose objects and is fundamental to probability and algebra.
In the exercise of arranging binary words of a given length, combinatorics enables the calculation of all possible combinations of binary digits. For instance, with three digits, each being able to take on the value of 0 or 1, there are precisely \(2^3\) or 8 possible unique combinations. Using combinatorics, we can solve problems related to the structure, enumeration, and arrangement of objects without having to list every single possibility, which becomes impractical with a larger number of bits. This branch of mathematics is incredibly powerful in computer science, information theory, and discrete mathematics as it provides the tools to handle vast quantities of data systematically.
In the exercise of arranging binary words of a given length, combinatorics enables the calculation of all possible combinations of binary digits. For instance, with three digits, each being able to take on the value of 0 or 1, there are precisely \(2^3\) or 8 possible unique combinations. Using combinatorics, we can solve problems related to the structure, enumeration, and arrangement of objects without having to list every single possibility, which becomes impractical with a larger number of bits. This branch of mathematics is incredibly powerful in computer science, information theory, and discrete mathematics as it provides the tools to handle vast quantities of data systematically.
Other exercises in this chapter
Problem 49
Let \(A, B,\) and \(C\) be subsets of a finite set \(U .\) Derive a formula for each. \(\left|A^{\prime} \cap B^{\prime} \cap C^{\prime}\right|\)
View solution Problem 49
Arrange the binary words of the given length in increasing order of magnitude. Length two.
View solution Problem 50
State the inclusion-exclusion principle for four finite sets \(A_{t}, 1 \leq\) \(i \leq 4 .\) (The formula contains 15 terms.)
View solution Problem 51
A ternary word is a word over the alphabet \(\\{0,1,2\\} .\) Arrange the ternary words of the given length in increasing order of magnitude. Length one.
View solution