# 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.

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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