anonymous
  • anonymous
Need help with a discrete math problem.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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anonymous
  • anonymous
1 Attachment
blockcolder
  • blockcolder
If \(\chi_A \cdot \chi_B=1\), then that means \(\chi_A=1\) and \(\chi_B=1\). If \(\chi_A \cdot \chi_B=0\), then that means one of the following is true: 1. \(\chi_A=0\), \(\chi_B=1\) 2. \(\chi_A=1\), \(\chi_B=0\) 3. \(\chi_A=\chi_B=0\).
blockcolder
  • blockcolder
Now, if \(\chi_A(x)=1\) and \(\chi_B(x)=1\), then this means \(x \in A\) and \(x \in B\). If \(\chi_A(x)=1\) and \(\chi_B(x)=0\), then \(x\in A\) and \(x \notin B\). If you repeat this for the rest, you should be able to figure out the set with characteristic function \(\chi_A \cdot \chi_B\)

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anonymous
  • anonymous
So does this mean XA*XB would only be in subset A?
anonymous
  • anonymous
@phi Do you know how to do this problem? I don't understand the solution =\
phi
  • phi
Do you have a definition for characteristic function ? I am guessing it means "is a member of " ?
anonymous
  • anonymous
So I just looked it up and found this: I(x) = { 1, if x belongs to B, = { 0, if x doesnot belong to B. where x is a variable, which can take any letter from the alphabets.
phi
  • phi
ok. then Xa * Xb would be 1 only if the member is in both sets
phi
  • phi
which sounds like A intersect B
anonymous
  • anonymous
ok
anonymous
  • anonymous
so it asks what subset of S has characteristic function (a), (b), and (c). would that mean a is (a) characteristic function of the set A?
phi
  • phi
(a) is the characteristic function of \( A \cap B\)
anonymous
  • anonymous
I don't understand how you got that =\
phi
  • phi
Xa (y) is 1 if y is in set A Xb (y) is 1 if y is in set B Xa (y) * Xb (y) is 1 if y is in set A and in set B by definition, if y is in A and in B it is in \( A \cap B\)
anonymous
  • anonymous
ohhhhhhhh makes sense!
anonymous
  • anonymous
so then then for part (b) if Xa + Xb - Xa(intersects)B = 1
phi
  • phi
I think this characteristic function is only allowed 0 or 1 as its output values Xa(y) + Xb(y) - \( X_{A \cap B}\)(y) y is in A but not in B = 1 + 0 - 0 = 1 y is in B but not in A = 0 +1 -0 = 1 y is in both A and B, then 1+1 - 1 = 1 y is not in A and not in B= 0 + 0 -0 =0
anonymous
  • anonymous
so I'm thinking something like A\B, B\A and A(union)B
phi
  • phi
It sure looks like A \( \cup \) B to me the reason the characteristic function is so complicated is that if y is in both A and B, just using Xa(y) + Xb(y) would give 2 , and that is not allowed.
anonymous
  • anonymous
so A\B and B\A wouldn't be answers?
phi
  • phi
what does A\B mean?
anonymous
  • anonymous
y is in A but not in B = 1 + 0 - 0 = 1 which is the Set A - the set B (A\B)
anonymous
  • anonymous
in A but not in B
anonymous
  • anonymous
oh wait, I don't think I understood it correctly
phi
  • phi
if you evaluate Xa(y) + Xb(y)-\(X_{A \cap B}\)(y) with y in B, you get a 1 but A\B should give you a 0 ?
anonymous
  • anonymous
ok, I think it makes sense now, so the answer would be AUB
anonymous
  • anonymous
because y is in both A and B
phi
  • phi
y is in one or the other or both
anonymous
  • anonymous
ah I see. Now it makes sense. :)
anonymous
  • anonymous
so for (c) y is in A but not in B = 1 + 0 - 2 * 0 = 1 y is in B but not in A = 0 + 1 - 2 * 0 = 1 y is in both A and B, then 1 + 1 - 2 * 1 = 0 y is not in A and not in B = 0 + 0 - 2 * 0 = 0
phi
  • phi
looks good.
anonymous
  • anonymous
So the answer would be A(compliment)(intersects)B(compliment)?
phi
  • phi
A' means not in A ? but y is in A but not in B = 1 + 0 - 2 * 0 = 1
anonymous
  • anonymous
oh right
anonymous
  • anonymous
hmm, I'm not sure
anonymous
  • anonymous
so when y is in both A and B it equals 0
phi
  • phi
I am thinking A \( \cup \) B - A \( \cap \) B
anonymous
  • anonymous
that would do it :O
anonymous
  • anonymous
Thank you so much for the help!
anonymous
  • anonymous
and for putting up with my stupidity :D
anonymous
  • anonymous
All makes sense now.

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