# Characteristic function and preimage?

Characteristic function and preimage???

## Homework Statement

Let S be a nonempty subset of ℝ.

Define χs= { 1 if x is in S and 0 if x is not in S

Determine χs-1(Q) [where Q=set of all rational numbers]

and χs-1((0,∞))

We haven't really dealt much with this function, and I really don't know how to go about doing this. I'm guessing for Q it will be all x in S such that f(x)=Q? Is that right?

Any help is appreciated,

Thanks.

HallsofIvy
Homework Helper

Then go back an review the definitions! Once you know the definitions, this problem is trivial.

Your function, "$X_S$" (strictly speaking, "$\chi_s$", the Greek letter "chi") is defined to be 1 if x is in S, 0 if not. In other words, the only possible value of x is 0 or 1, both of which are rational numbers, but only 1 is in $(0, \infty)$.

Thank you, so for the rationals, the preimage of chi sub s is the set of all numbers in S and the complement of S, i.e. the set of all reals. Is that correct?

SammyS
Staff Emeritus
Homework Helper
Gold Member

Thank you, so for the rationals, the preimage of chi sub s is the set of all numbers in S [STRIKE]and[/STRIKE] or in the complement of S, i.e. the set of all reals. Is that correct?
Yes.

(There are no numbers which are both in set S and in the compliment of set S.)