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 2 years 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?
 2 years 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?

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Nodata
 2 years ago
Best ResponseYou've already chosen the best response.0I thought that it would be (11  5) + (12  5) = 13

Nodata
 2 years ago
Best ResponseYou've already chosen the best response.0I'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.

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

Nodata
 2 years ago
Best ResponseYou've already chosen the best response.0Can you see the math symbols? I think because I can't =/

aacehm
 2 years ago
Best ResponseYou've already chosen the best response.0well, 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.

aacehm
 2 years ago
Best ResponseYou've already chosen the best response.0If they aren't disjoint, your solution of 13 sounds right.

Nodata
 2 years ago
Best ResponseYou've already chosen the best response.0@aacehm You mean \( #(A \cap B) + #(A \cap C) \)

aacehm
 2 years ago
Best ResponseYou've already chosen the best response.0Yeah, but the number signs aren't working.

Nodata
 2 years ago
Best ResponseYou've already chosen the best response.0Is this equality right? \( \#A\cap(B\cap C) = \#A + \# (B\cap C) \)

Nodata
 2 years ago
Best ResponseYou've already chosen the best response.0This is so confusing =/

aacehm
 2 years ago
Best ResponseYou've already chosen the best response.0That 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

aacehm
 2 years ago
Best ResponseYou've already chosen the best response.0is there information given about which sets are disjoint?

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

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

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

Nodata
 2 years ago
Best ResponseYou've already chosen the best response.0Then as you said @mathmate the miminum cardinality of A is 18

mathmate
 2 years ago
Best ResponseYou've already chosen the best response.1Yep, that's very convincing! Good job!
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