Example 1.2.5(a):
Let f(x) = 0 if x is rational and f(x) = 1 if x
is irrational. This function is called Dirichlet’s Function. The range for
f is R. Find the image of the domain of the Dirichlet Function when:
- the domain of f is Q
- the domain of f is R
- the domain of f is [0, 1] (the closed interval between 0 and 1)
- When the domain is Q, we have that f(x) = 0 for any x,
because x must be a rational number. Hence, the image of the
domain Q is the set consisting of the single element {0}.
- When the domain is R, we have that f(x) could be 0 or 1,
because x could be rational or irrational. f(x) can not be
any other number. Hence, the image of the domain R is the set consisting
of the two elements {0, 1}.
- The interval [0, 1] contains irrationals as well as rational numbers. Therefore, f(x) could be equal to 0 or 1, and the image of [0, 1] under f is the set with the two elements {0, 1}.