Find the domain and range of these functions. a) the function that assigns to each pair of po

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 ${\displaystyle N-\left\{0\right\}}$
${\displaystyle Domain=\{(x,y)\mid x=1,2,3,\dots \text{and}y=1,2,3,\dots \}}$
${\displaystyle =\{(x,y)\mid x\in N-\left\{0\right\}\wedge y\in N-\left\{0\right\}\}}$
${\displaystyle =(N-\left\{0\right\})\times (N-\left\{0\right)\})}$
The range is the first positive integer in the pair in the domain.
${\displaystyle Range=\{1,2,3,\dots \}=N-\left\{0\right\}}$
b)We have the function that assigns to each positive integer its largest decimal digit
The domain is the set of all positive integers
${\displaystyle Domain=\{1,2,3,\dots \}=N-\left\{0\right\}}$
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
${\displaystyle Range=\{1,2,3,4,5,6,7,8,9\}}$
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
${\displaystyle Domain=\{\lambda ,0,1,00,01,11,10,010,011,\dots \}}$
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
${\displaystyle Range=\{\dots ,-2,-1,0,1,2,3,\dots \}}$
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:
${\displaystyle Domain=\{1,2,3,\dots \}=N-\left\{0\right\}}$
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,
${\displaystyle Range=\{1,2,3,4,\dots \}=N-\left\{0\right\}}$
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
${\displaystyle Domain=\{\lambda ,0,01,11,10,010,\dots \}}$
The range is the set of all longest strings of ones in any string, Hence, the range contains only string containing the digit 1
${\displaystyle Range=\{\lambda ,1,11,111,1111,11111,\dots .\}}$