Arthur Pratt

2022-01-06

Determine the domain and range of these functions.
a) The function that assigns to each pair of positive integers the first integer of the pai is the function that assigns the first element of each pair to the pair.
b) The function that assigns the largest decimal digit to each positive integer.
c) the function that assigns to a bit string the number of ones minus the number of zeros in the string.
d) the function that assigns to each positive integer the largest integer not exceeding the square root of the integer.
e) the function that assigns to a bit string the longest string of ones in the string.

Bob Huerta

I'll do the first 3 for the example:
a)We have a function that assigns to each pair of positive integers the first integer of the pair
The domain is set of all pairs of positive integers. A positive integer is a natural number and 0 is the only natural number that is'nt positive. This implies that the group of positive integers is $N-\left\{0\right\}$

$=\left\{\left(x,y\right)\mid x\in N-\left\{0\right\}\wedge y\in N-\left\{0\right\}\right\}$
$=\left(N-\left\{0\right\}\right)×\left(N-\left\{0\right)\right\}\right)$
The range is the first positive integer in the pair in the domain.
$Range=\left\{1,2,3,\dots \right\}=N-\left\{0\right\}$
b)We have the function that assigns to each positive integer its largest decimal digit
The domain is the set of all positive integers
$Domain=\left\{1,2,3,\dots \right\}=N-\left\{0\right\}$
Since the domain contains only positive integers and doesn't have 0, the range doesn't contain 0. What is more, the digits are the values from 1 to 9, so
$Range=\left\{1,2,3,4,5,6,7,8,9\right\}$
c)Given the function that assigns to a bit string the number of ones minus the number of zeros in the string
The domain is set of all bit strings
$Domain=\left\{\lambda ,0,1,00,01,11,10,010,011,\dots \right\}$
As the range is the set of all differences between the number of ones and number of zeros in a string, it can take on negative values, on zero and on positive values. Thus, we have the range
$Range=\left\{\dots ,-2,-1,0,1,2,3,\dots \right\}$
d)We have a function that assigns to each positive integer the largest integer not exceeding the square root of the integer
The domain is the set of all positive integers:
$Domain=\left\{1,2,3,\dots \right\}=N-\left\{0\right\}$
The range is the set of the largest integer not exceeding the square root of a positive integer, for example the image of 1 is 1(and the largest integer not exceeding 1 is 1)
We can note that all positive integers are contained in the set, thus,
$Range=\left\{1,2,3,4,\dots \right\}=N-\left\{0\right\}$
e)Finally, we have the function that assigns to a bit string the longest string of obnes in the string
The domain is the set of all bit strings
$Domain=\left\{\lambda ,0,01,11,10,010,\dots \right\}$
The range is the set of all longest strings of ones in any string, Hence, the range contains only string containing the digit 1
$Range=\left\{\lambda ,1,11,111,1111,11111,\dots .\right\}$

karton

a) The assignment function for each pair of positive integers The pair's first integer can be written as follows:
$f:{ℤ}^{+}×{ℤ}^{+}\to {ℤ}^{+}$
$f\left(x,y\right)=x$
The domain of this function is the set of all pairs of positive integers, denoted as ${ℤ}^{+}×{ℤ}^{+}$. The range of the function is the set of positive integers, denoted as ${ℤ}^{+}$.
b) The function that assigns the largest decimal digit to each positive integer can be represented as:
$g:{ℤ}^{+}\to \left\{0,1,2,3,4,5,6,7,8,9\right\}$

The domain of this function is the set of positive integers, denoted as ${ℤ}^{+}$. The range of the function is the set of decimal digits, specifically the set $\left\{0,1,2,3,4,5,6,7,8,9\right\}$.
c) The function that assigns to a bit string the number of ones minus the number of zeros in the string can be represented as:
$h:\left\{0,1{\right\}}^{*}\to ℤ$
$h\left(s\right)=\text{count}\left(s,1\right)-\text{count}\left(s,0\right)$
Here, $\left\{0,1{\right\}}^{*}$ represents the set of all possible bit strings, including the empty string. The domain of this function is the set of bit strings, denoted as $\left\{0,1{\right\}}^{*}$. The range of the function is the set of integers, denoted as $ℤ$.
d) The function that assigns to each positive integer the largest integer not exceeding the square root of the integer can be represented as:
$k:{ℤ}^{+}\to ℤ$
$k\left(n\right)=⌊\sqrt{n}⌋$
The domain of this function is the set of positive integers, denoted as ${ℤ}^{+}$. The range of the function is the set of integers, denoted as $ℤ$.
e) The function that assigns to a bit string the longest string of ones in the string can be represented as:
$m:\left\{0,1{\right\}}^{*}\to ℕ$

The domain of this function is the set of bit strings, denoted as $\left\{0,1{\right\}}^{*}$. The range of the function is the set of non-negative integers, denoted as $ℕ$.

alenahelenash

Step a) The domain and range of the function that assigns to each pair of positive integers the first integer of the pair can be determined as follows:
Domain: $D=\left\{\left(x,y\right)\mid x,y\in {\mathbb{Z}}^{\mathbb{+}}\right\}$
Range: $R=\left\{x\mid x\in {\mathbb{Z}}^{\mathbb{+}}\right\}$
Step b) The domain and range of the function that assigns the largest decimal digit to each positive integer can be determined as follows:
Domain: $D=\left\{x\mid x\in {\mathbb{Z}}^{\mathbb{+}}\right\}$
Range: $R=\left\{0,1,2,3,4,5,6,7,8,9\right\}$
Step c) The domain and range of the function that assigns to a bit string the number of ones minus the number of zeros in the string can be determined as follows:
Domain:
Range: $R=\left\{x\mid x\in ℤ\right\}$
Step d) The domain and range of the function that assigns to each positive integer the largest integer not exceeding the square root of the integer can be determined as follows:
Domain: $D=\left\{x\mid x\in {\mathbb{Z}}^{\mathbb{+}}\right\}$
Range: $R=\left\{y\mid y\in {\mathbb{Z}}^{\mathbb{+}},y\le ⌊\sqrt{x}⌋\right\}$
Step e) The domain and range of the function that assigns to a bit string the longest string of ones in the string can be determined as follows:
Domain:
Range:

star233

a) The function that assigns to each pair of positive integers the first integer of the pair is the function that assigns the first element of each pair to the pair.
The domain of this function is the set of all pairs of positive integers, denoted by ${ℤ}^{+}×{ℤ}^{+}$. The range of the function is the set of all positive integers, denoted by ${ℤ}^{+}$.
b) The function that assigns the largest decimal digit to each positive integer.
The domain of this function is the set of positive integers, denoted by ${ℤ}^{+}$. The range of the function is the set of decimal digits from 0 to 9, denoted by $\left\{0,1,2,3,4,5,6,7,8,9\right\}$.
c) The function that assigns to a bit string the number of ones minus the number of zeros in the string.
The domain of this function is the set of all bit strings, denoted by $\left\{0,1{\right\}}^{*}$. The range of the function is the set of integers, as the result can be positive, negative, or zero. We can denote the range as $ℤ$.
d) The function that assigns to each positive integer the largest integer not exceeding the square root of the integer.
The domain of this function is the set of positive integers, denoted by ${ℤ}^{+}$. The range of the function is the set of integers that are less than or equal to the largest integer not exceeding the square root of any positive integer. We can denote the range as $⌊\sqrt{n}⌋$ for any positive integer $n$.
e) The function that assigns to a bit string the longest string of ones in the string.
The domain of this function is the set of all bit strings, denoted by $\left\{0,1{\right\}}^{*}$. The range of the function is the set of non-negative integers, representing the length of the longest string of ones in the given bit string. We can denote the range as ${ℤ}_{\ge 0}$.

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