Question

Find the domain and range of these functions. a) the function that assigns to each pair of positive inte- gers the maximum of these two integers b) the function that assigns to each positive integer the number of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 that do not appear as decimal digits of the integer c) the function that assigns to a bit string the number of times the block 11 appears d) the function that assigns to a bit string the numerical position of the first 1 in the string and that assigns the value 0 to a bit string consisting of all Os

Answer 1

Answer:

See below

Step-by-step explanation:

a)

Domain: the cartesian product of integers ZxZ

Range: the integers Z, for every p in Z is the image of (p, p-1)

b)

Domain: the positive integers Z+

Range: the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} obviously, f(1)=9, f(12)=8, f(123)=7,...,f(123456789)=1 and f(1234567890)=0. So, every number in {0,1,...,9} is the image of some positive integer.

c)

Domain: all the bit strings. Since any real number can be written as a bit string in the binary system, the domain is actually the set R of real numbers.

Range: a bit string can have 0 or 1 or 2 or more blocks “11”, so the range is the set N of natural numbers including 0.

d)

Domain: all the bit strings (the real numbers as in c).  

Range: the natural numbers N plus 0