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 digits0, 1, 2, 3, 4, 5, 6, 7, 8, 9 that do not appear as decimal digits of theinteger 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 0s
Short Answer
a) Domain: ℕ x ℕ, Range: ℕ. b) Domain: ℕ, Range: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. c) Domain: bit strings, Range: ℕ0. d) Domain: bit strings, Range: ℕ0.
Step by step solution
01
Title - Determine the domain and range of function a
Consider the function that assigns to each pair of positive integers the maximum of these two integers. The domain of this function is all pairs of positive integers (ℕ x ℕ). The range is the set of all positive integers (ℕ) because the maximum of any two positive integers is always a positive integer.
02
Title - Determine the domain and range of function b
Consider the function that assigns to each positive integer the number of decimal digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 that do not appear in the integer. The domain of this function is the set of all positive integers (ℕ). The range is the set of integers from 0 to 10, inclusive, as the minimum value is 0 (all digits appear) and the max is 10 (no digits appear).
03
Title - Determine the domain and range of function c
Consider the function that assigns to a bit string the number of times the block 11 appears. The domain is the set of all bit strings. The range is the set of non-negative integers (ℕ0) because the count of '11' appearances can range from 0 to the length of the bit string/2.
04
Title - Determine the domain and range of function d
Consider the function that assigns to a bit string the numerical position of the first 1 in the string and assigns the value 0 to a bit string consisting entirely of 0s. The domain is the set of all bit strings. The range is the set of non-negative integers (ℕ0) where each non-zero output indicates the position of the first '1', and 0 if no '1's are present in the string.
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Domain of a Function
When we talk about the 'domain of a function,' we are referring to the set of all possible input values that the function can accept.
In mathematical terms, if we have a function denoted by \(f(x)\), then the domain includes all the values of \(x\) for which \(f(x)\) is defined.
For example, consider the function in part (a) of the exercise. This function assigns to each pair of positive integers the maximum of the two integers.
The domain here is all pairs of positive integers, which we denote as \( \mathbb{N} \times \mathbb{N} \).
Range of a Function
The 'range of a function' is the set of all possible output values that a function can produce, given its domain.
In simple terms, if you feed all possible inputs from the domain into the function, the range is the set of outputs you get.
Let's take the function in part (b), which assigns to each positive integer the number of decimal digits (0, 1, 2, ..., 9) that do not appear in the integer.
The range for this function is all integers from 0 to 10 because, in the best case, all digits are present (yielding 0), and in the worst case, no digits are present (yielding 10).
Positive Integers
Positive integers are the set of whole numbers greater than zero. They are denoted by \( \mathbb{N} \) and include {1, 2, 3, 4, ...}.
They do not include zero or any negative numbers.
For example, in part (b) of the exercise, the domain is the set of all positive integers because the function deals with positive integers only. Similarly, in part (a), the function also operates on pairs of positive integers, making \( \mathbb{N} \times \mathbb{N} \) its domain.
Bit Strings
A bit string is a sequence of bits, where each bit is either 0 or 1. Bit strings are used extensively in computer science.
They represent data and perform various computational tasks.
For example, consider the function in part (c) of the exercise, which counts the number of times the block '11' appears in a bit string.
The domain here is the set of all possible bit strings, and the range is the set of non-negative integers (\( \mathbb{N}_{0} \)), as it counts occurrences from 0 upwards.
Another example is part (d), where the function returns the position of the first '1' in the bit string or 0 if no '1' is present.
This function, too, has all bit strings as its domain and non-negative integers as its range.
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