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Cardinality properties:
Suppose you have three sets A, B, and C, satisfying the following conditions:
\( \# (A\cap B)=11 \)
\( \# (A\cap C)=12\)
\( \# (A\cap B \cap C)=5\)
What is the minimun cardinality of set A?
 one year ago
 one year ago
Cardinality properties: Suppose you have three sets A, B, and C, satisfying the following conditions: \( \# (A\cap B)=11 \) \( \# (A\cap C)=12\) \( \# (A\cap B \cap C)=5\) What is the minimun cardinality of set A?
 one year ago
 one year ago

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NodataBest ResponseYou've already chosen the best response.0
I thought that it would be (11  5) + (12  5) = 13
 one year ago

NodataBest ResponseYou've already chosen the best response.0
I'm not sure if I'm right and I don't know if the fact that B a C are disjoint set could afect the cardinality.
 one year ago

NodataBest ResponseYou've already chosen the best response.0
And also I don't know if I can figure out the maximun cardinality with this information. Or even the cardinality of set A.
 one year ago

NodataBest ResponseYou've already chosen the best response.0
Can you see the math symbols? I think because I can't =/
 one year ago

aacehmBest ResponseYou've already chosen the best response.0
well, if B and C are disjoint, wouldn't it make sense that the cardinality of A is at least \[(A \cap B) + (A \cap C)\]? Because the minimum cardinality of B has to be 11, which means that A is at least 11, but there are also 12 elements that are not in B but have to be in A as well.
 one year ago

aacehmBest ResponseYou've already chosen the best response.0
If they aren't disjoint, your solution of 13 sounds right.
 one year ago

NodataBest ResponseYou've already chosen the best response.0
@aacehm You mean \( #(A \cap B) + #(A \cap C) \)
 one year ago

aacehmBest ResponseYou've already chosen the best response.0
Yeah, but the number signs aren't working.
 one year ago

NodataBest ResponseYou've already chosen the best response.0
Is this equality right? \( \#A\cap(B\cap C) = \#A + \# (B\cap C) \)
 one year ago

NodataBest ResponseYou've already chosen the best response.0
This is so confusing =/
 one year ago

aacehmBest ResponseYou've already chosen the best response.0
That equality doesn't sound right.\[A = \left\{5, 6, 7..100\right\}\] \[B = \left\{1, 2, 3\right\}\] \[C = \left\{3,4,5\right\}\] Then the left side of your equation would 1, and the right side would be 97
 one year ago

aacehmBest ResponseYou've already chosen the best response.0
is there information given about which sets are disjoint?
 one year ago

aacehmBest ResponseYou've already chosen the best response.0
Actually, from statement 3, you can infer that B and C aren't disjoint, because if they were, then it would equal 23, not 5
 one year ago

mathmateBest ResponseYou've already chosen the best response.1
The minimum cardinality of A is \( \#(A\cap B)\cup (A \cap C)=\#(A\cap B) +\#(A \cap C)  \#(A\cap B \cap C)=11+125=18 \)
 one year ago

mathmateBest ResponseYou've already chosen the best response.1
dw:1356902619299:dw
 one year ago

NodataBest ResponseYou've already chosen the best response.0
Thank you, this is what i got: \( A\cap(B\cup C)\subseteq A\) Then \(\#A\geq \#[ A\cap(B\cup C)]\) \(\#[ A\cap(B\cup C)]=\#[(A\cap B)\cup(A\cap C)]\) \(\#[ A\cap(B\cup C)]=\#(A\cap B)+\#(A\cap C)\#(A\cap\ B\cap C)\) \(\#[ A\cap(B\cup C)]=11+125=18\)
 one year ago

NodataBest ResponseYou've already chosen the best response.0
Then as you said @mathmate the miminum cardinality of A is 18
 one year ago

mathmateBest ResponseYou've already chosen the best response.1
Yep, that's very convincing! Good job!
 one year ago
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