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

Beginner2022-01-07Added 41 answers

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 $N-\left\{0\right\}$

$Domain=\{(x,y)\mid x=1,2,3,\dots {\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}y=1,2,3,\dots \}$

$=\{(x,y)\mid x\in N-\left\{0\right\}\wedge y\in N-\left\{0\right\}\}$

$=(N-\left\{0\right\})\times (N-\left\{0\right)\})$

The range is the first positive integer in the pair in the domain.

$Range=\{1,2,3,\dots \}=N-\left\{0\right\}$

b)We have the function that assigns to each positive integer its largest decimal digit

The domain is the set of all positive integers

$Domain=\{1,2,3,\dots \}=N-\left\{0\right\}$

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

$Range=\{1,2,3,4,5,6,7,8,9\}$

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

$Domain=\{\lambda ,0,1,00,01,11,10,010,011,\dots \}$

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

$Range=\{\dots ,-2,-1,0,1,2,3,\dots \}$

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:

$Domain=\{1,2,3,\dots \}=N-\left\{0\right\}$

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,

$Range=\{1,2,3,4,\dots \}=N-\left\{0\right\}$

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

$Domain=\{\lambda ,0,01,11,10,010,\dots \}$

The range is the set of all longest strings of ones in any string, Hence, the range contains only string containing the digit 1

$Range=\{\lambda ,1,11,111,1111,11111,\dots .\}$

karton

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