Question
Suppose n > 1 is a natural number and f : Z --> N U {0} is the function that associates with each a E Z its remainder upon division by n; thus if a = qn + r with 0 less than or equal to r less than n, then f(a) = r. a) find the domain and range of f
Suppose n > 1 is a natural number and f : Z --> N U {0} is the function that associates with each a E Z its remainder upon division by n; thus if a = qn + r with 0 less than or equal to r less than n, then f(a) = r. a) find the domain and range of f
Explanation / Answer
f: /doubleZ -> /doubleN /union {0} Whenever you define a function this way, the set on the left of the arrow is the domain, and the set on the right is the codomain. So the domain is /doubleZ. (/doubleZ is the Z written with the double slant, representing the set of integers) The range is the set of all values in the codomain that are the image of something in your domain. For any a in doubleZ, this mapping will only give you 0, 1, 2,... N-1. So the range is {0, 1, 2, ... N-1}. Remark. This function can be viewed as a homomorphism from the integers onto residue classes modulo N. It's the homomorphism that gives rise to the quotient group (or field if you prefer) /doubleZ N /doubleZ, or the integers factored by the normal subgroup N /doubleZ .