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Functions
Exercise 1. Find the domain and range of these functions. Explain.
a) the function that assigns to each negative integer its last digit; The domain is the set of all negative integers Zโ^ = {โ 1 , โ 2 , โ 3 ,... , }. The range is the set of all digits, namely, { 0 , 1 , 2 ,... , 8 , 9 }.
b) the function that assigns the next largest integer to a positive integer; The domain is the set of all positive integers { 1 , 2 , 3 , 4 ,... , }. The range is the set of all positive integers larger than or equal to 2, namely, { 2 , 3 , 4 ,... , }.
b) the function that assigns the SSN to each eligible resident of US. The domain is the set of all US residents who are eligible to get the social security number. The range is the set of the nine-digit numbers which are used as SSN.
Exercise 2. Let f is given by formula f (x) = 1/(x + 1). Determine whether f is a function or not. If it is, find its domain, codomain, range, determine whether this function is surjective, injective, bijective, and find its inverse, if it exists. Explain.
a) f : Z โ R; It is not a function because it is not defined at x = โ 1 ;
b) f : R โ R; It is not a function because it is not defined at x = โ 1 ;
c) f : R \ {โ 1 } โ R; It is a function. It is injective but not surjective. For instance, 0 โ R but f (x) 6 = 0 for all x from the domain.
d) f : R \ {โ 1 } โ R \ { 0 }. It is a bijective function. The inverse function is given by formula f โ^1 (y) = โ1 + 1/y.
Exercise 3. Let f (x) = ax + b and g(x) = cx + d be two functions from R to R, where a, b, c, and d are constants. Determine for which constants a, b, c, and d it is true that f โฆ g = g โฆ f.
In order to solve this problem, we simply compute the functions f โฆ g and g โฆ f.
(f โฆg)(x) = f (g(x))(x) = a(g(x))+b = a(cx+d) + b = (ac)x+(ad+b),
(g โฆf )(x) = g(f (x))(x) = c(f (x))+d = c(ax+b) + d = (ac)x+(bc+d).
From the above formulas, we see that the function are the same iff
ad + b = bc + d.
Exercise 4. Determine whether each of these sets is countable or uncount- able. For those that countable, exhibit a one-to-one correspondence between N (or some finite subset of N) and that set. For those that are not countable, show that such bijection does not exist (adapt a proof from the book).
a) the negative integers; The mapping from N to this set is defined by f (n) = โn โ 1.
b) the odd negative integers; The mapping from N to this set is defined by f (n) = โ 2 n โ 1.
c) all real numbers between 0 and 1; This set is not countable. The proof is given in the book.
d) all integers between 1 and 1024. This set is finite, and its cardinal number (cardinality) is equal to 1024. Also, this set is a subset of N. Therefore, we can choose the identity mapping i(n) = n as the desired bijection.
