Find the domain and range of these functions. a) the function that assigns to each pair of positive integers the ﬁrst integer of the pair b) the function that assigns to each positive integer its largest decimal digit 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

a) The function that assigns to each pair of positive integers the first integer of the pair will have domain equal to all pairs of natural numbers i.e, N x N where N = {1,2,3,.....} since the set of natural numbers has all positive integers.

Domain = {(x,y) | x ∈ N and y ∈ N}

The range is any integer so it will be a set of natural numbers N = {1,2,3,....}

b) Here, we have domain of each positive integer so we will have

Domain = {1,2,3,....}

Range is the largest decimal digit of the positive number which can be any number from 1 to 9

So, Range = {1,2,3,4,....,9}

c) Domain is a bit string i.e.

Domain = {0, 1, 00, 01, 10, 11,...}

The range includes the differences of 1's and 0's in the string. According to this, we can have all positive and non-positive (0 and negative integers) values in the range. So, it will be a set of all integers

Range = {.....,-4, -3, -2, -1, 0, 1, 2, 3,......}

d) We again have domain equal to set of positive integers i.e,

Domain = {1,2,3,4,.....}

Range is the largest integer that does not exceed the square root of the number. According to this, image of 1 will be 0, image of 2 will be 1. So, range will be a set of whole numbers.

Range = {0, 1, 2, 3, 4,.....}

e) Domain = {0, 1, 01, 10, 11,....} (A bit string)

Longest string of ones in the string will give the range

Range = {1, 11, 111, 1111, 11111,.....}