Characteristic function and preimage?

[SMA_01]

Characteristic function and preimage?

Homework Statement

Let S be a nonempty subset of ℝ.

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

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

and χ_s^-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]

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 $(0, \infty)$.

 

[SMA_01]

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]

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.)