## No-data 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?

1. No-data

I thought that it would be (11 - 5) + (12 - 5) = 13

2. No-data

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.

3. No-data

And also I don't know if I can figure out the maximun cardinality with this information. Or even the cardinality of set A.

4. No-data

Can you see the math symbols? I think because I can't =/

5. aacehm

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.

6. aacehm

If they aren't disjoint, your solution of 13 sounds right.

7. No-data

@aacehm You mean $$#(A \cap B) + #(A \cap C)$$

8. aacehm

Yeah, but the number signs aren't working.

9. aacehm

sorry

10. No-data

Is this equality right? $$\#A\cap(B\cap C) = \#A + \# (B\cap C)$$

11. No-data

This is so confusing =/

12. aacehm

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

13. aacehm

is there information given about which sets are disjoint?

14. aacehm

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

15. mathmate

The minimum cardinality of A is $$\#(A\cap B)\cup (A \cap C)=\#(A\cap B) +\#(A \cap C) - \#(A\cap B \cap C)=11+12-5=18$$

16. mathmate

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17. No-data

Thank you, this is what i got: $$A\cap(B\cup C)\subseteq A$$ Then $$\#A\geq \#[ A\cap(B\cup C)]$$ $$\#[ A\cap(B\cup C)]=\#[(A\cap B)\cup(A\cap C)]$$ $$\#[ A\cap(B\cup C)]=\#(A\cap B)+\#(A\cap C)-\#(A\cap\ B\cap C)$$ $$\#[ A\cap(B\cup C)]=11+12-5=18$$

18. No-data

Then as you said @mathmate the miminum cardinality of A is 18

19. mathmate

Yep, that's very convincing! Good job!