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Characteristic function and preimage?

  • Thread starter SMA_01
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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.
 

Answers and Replies

  • #2
HallsofIvy
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Then go back an review the definitions! Once you know the definitions, this problem is trivial.

Your function, "[itex]X_S[/itex]" (strictly speaking, "[itex]\chi_s[/itex]", 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 [itex](0, \infty)[/itex].
 
  • #3
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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?
 
  • #4
SammyS
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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.)
 

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