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Chapter 2: Problem 7

Find the domain and range of these functions. a) the function that assigns to each pair of positive integers the maximum ofthese two integers b) the function that assigns to each positive integer the number of the digits\(0,1,2,3,4,5,6,7,8,9\) that do not appear as decimal digits of the integer c) the function that assigns to a bit string the number of times the block 11appears d) the function that assigns to a bit string the numerical position of thefirst 1 in the string and that assigns the value 0 to a bit string consistingof all 0 s

### Short Answer

Expert verified

a) Domain: \( \{ (m, n) \mid m > 0, n > 0 \} \), Range: \( \{ n \mid n \, \in \, \mathbb{Z}^+ \} \). b) Domain: \( \{ n \mid n > 0 \} \), Range: \( \{ k \mid 0 \leq k \leq 10 \} \). c) Domain: \( \{ s \mid s \text{ is a bit string} \} \), Range: \( \{ k \mid k \in \mathbb{N}\} \). d) Domain: \( \{ s \mid s \text{ is a bit string} \} \), Range: \( \{ 0 \} \cup \{ k \mid k \in \mathbb{Z}^+, k \leq n \} \).

## Step by step solution

01

## Part a: Finding the Domain

The function takes a pair of positive integers as input. Since we are looking at all pairs of positive integers, the domain is all possible pairs of positive integers: Domain: \( \{ (m, n) \mid m > 0, n > 0 \} \).

02

## Part a: Finding the Range

The function returns the maximum of the two integers. Given that both integers are positive, the minimum value returned is 1 and there is no upper limit. So, the range is all positive integers: Range: \( \{ n \mid n \, \in \, \mathbb{Z}^+ \} \).

03

## Part b: Finding the Domain

The function assigns a value to each positive integer, so the domain is all positive integers: Domain: \( \{ n \mid n > 0 \} \).

04

## Part b: Finding the Range

A positive integer can have decimal digits from 0 to 9. It is possible that none or all of the digits can be absent from the number. Therefore, the range (number of missing digits) can be from 0 to 10: Range: \( \{ k \mid 0 \leq k \leq 10 \} \).

05

## Part c: Finding the Domain

The function takes any bit string as input, and a bit string can be any sequence of 0s and 1s, including the empty string: Domain: \( \{ s \mid s \text{ is a bit string} \} \).

06

## Part c: Finding the Range

The function counts the number of times the block '11' appears in the bit string. Depending on the length and content of the string, this number can range from 0 to \(\lfloor \frac{n}{2} \rfloor\) for a string of length \(n\): Range: \( \{ k \mid k \in \mathbb{N}\} \).

07

## Part d: Finding the Domain

The function takes any bit string as input, and a bit string can be any sequence of 0s and 1s: Domain: \( \{ s \mid s \text{ is a bit string} \} \).

08

## Part d: Finding the Range

The function returns the numerical position of the first 1 in the string or 0 if the string is all zeros. Hence, the possible values are 0 or any positive integer less than or equal to the length of the string: Range: \( \{ 0 \} \cup \{ k \mid k \in \mathbb{Z}^+, k \leq n \} \).

## Key Concepts

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

###### functions in discrete mathematics

In discrete mathematics, a function is an important concept that describes a specific relationship between two sets. A function assigns every element of one set, called the domain, to exactly one element of another set, called the range. There are different types of functions which serve various purposes, such as those involving integers and bit strings. Understanding functions is crucial for solving problems in discrete mathematics as they help in defining rules and structures within mathematical systems.

For example, if we have a function that assigns to each positive integer the number of unique digits it has, the domain is the set of all positive integers, and the range is the count of distinct digits from 0 to 9.

###### domain and range

Domains and ranges are foundational concepts when dealing with functions in discrete mathematics.

The domain is the complete set of possible input values for the function. In other words, it is the set of all values that you can put into the function. For instance, the domain of a function that takes a pair of positive integers is all ordered pairs of positive integers.

The range, on the other hand, is the complete set of possible output values which result from using the function. It consists of all the values that the function can output given the domain. For a function that assigns the maximum of two positive integers, the range would be all positive integers since any positive integer can be a maximum.

- The domain and range help in understanding the scope and limits of functions.
- They define what inputs are acceptable and what outputs are possible.
- Properly identifying domain and range is essential for accurately solving problems involving functions.

###### positive integers

Positive integers are the set of all whole numbers greater than zero. These numbers are denoted as \(\text{1, 2, 3, 4, ...}\), and they are crucial in many mathematical problems and functions.

In the context of discrete mathematics, positive integers often make up the domain of functions because they provide a clear, countable set. For example, in a function that takes a positive integer and counts the missing digits between 0 and 9, the domain is all positive integers, and the range of the function's possible outputs ranges from 0 to 10.

Positive integers are important because they define quantities and positions in many different mathematical scenarios. They allow for precise calculations and are used in the set theory, number theory, and other areas of mathematics.

###### bit strings

Bit strings are sequences of bits (0s and 1s) and are widely used in computer science and discrete mathematics. A bit string can represent data, instructions, or any form of binary-encoded information.

In the provided exercise, different functions involve bit strings in interesting ways, such as counting occurrences of specific patterns like '11' or finding the position of the first '1' in the string.

Bit strings can be of any length, including an empty string, making them a versatile input for functions.

- Domains involving bit strings include all possible combinations of 0s and 1s.
- They help in studying computational problems and algorithms.
- Understanding bit strings and how to manipulate them is essential for fields like cryptography, data compression, and network communication.

For instance, a function that counts the blocks '11' in a bit string will have a domain of all bit strings and a range based on the length and content of the bit string.

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