Finding the cardinal number for the intersection of two sets

  • #1
JC2000
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Homework Statement
A survey shows that ##63 %## Americans like cheese where as ##76 %## like apples. If ##x %## like both, find ##x##.
Relevant Equations
Since ##A \cap B \subset A## and ## A \cap B \subset B## :
##n(A \cap B) \leq n(A)## and ##n(A \cap B) \leq n(B)##
i.e ##n(A \cap B) \leq 63##
Also, ##n(A\cap B ) \geq 39## since ##n(A\cap B) = n(A)+ n(B)-n(A\cup B)## but ##n(A\cup B) \leq 100##

Thus : ## 39 \leq x \leq 63##
My Question :

1.Why are the inequalities considered? Why not simply use ##n(A\cap B) = n(A)+ n(B)-n(A\cup B)## to get ## n(A\cap B) = 39## ?
2. The way I interpret this is : If the set for people liking cheese was to be a subset of the set for people who like apples then the most number of people to like both would be 63. But I still fail to understand why the minimum value should be 39 (Why : ##P(A) + P(B) - P(A \cup B) < 1##)?
 
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  • #2
JC2000 said:
My Question :

1.Why are the inequalities considered? Why not simply use ##n(A\cap B) = n(A)+ n(B)-n(A\cup B)## to get ## n(A\cap B) = 39## ?
Because it's not given how many like both cheese and apples.
 
  • #3
Mark44 said:
Because it's not given how many like both cheese and apples.

Yes but we are given the remaining variables from which ##n(A \cap B)## can be found (?).
 
  • #4
JC2000 said:
Yes but we are given the remaining variables from which ##n(A \cap B)## can be found (?).
No, since you aren't given ##n(A \cup B)##
 
  • #5
Mark44 said:
No, since you aren't given ##n(A \cup B)##
Can't it be assumed to be 100? (Also could you shed some light on Q2?) Thanks!
 
  • #6
JC2000 said:
Can't it be assumed to be 100? (Also could you shed some light on Q2?) Thanks!
All you are given is that ##n(A \cup B) \le 100##, which doesn't imply that it equals 100.
I need to take off in a bit, so maybe somebody else can take a crack at your other question.
 
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  • #7
JC2000: I assume your numbers are given as percentages, right, so that it should be 76% and 63%?Edit. As Mark44 wrote, you don't have all the data you need. But you are correct that the percentage that like both is at most 63( Since it is a subset of those who like cheese) and the percentage that likes either is at most 100
 
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  • #8
Oh yes! Thanks!
 
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  • #9
As Mark44 pointed out, notice that ##n(A\cup B) ## is _at most_ 100, but not necessarily 100. Can you see why?
 
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  • #10
WWGD said:
As Mark44 pointed out, notice that ##n(A\cup B) ## is _at most_ 100, but not necessarily 100. Can you see why?
Is it because there may be a percentage of people that like neither? So this would mean ##n(A) + n(B)-n(A\cap B) \leq n(A\cup B)##?
 
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  • #11
Correct. Good job.
 
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  • #12
Thanks a lot!
 
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