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November 14th, 2018, 09:22 AM   #1
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Number of elements in set

A N B is not empty set
A U B has 17 elements
Number of subsets of A N B = n (A - B)

Theb what is the minimum number of elements of B?

Answer is 9 but i can not get there

I drew Venn diagram and said a+b+c = 17 (these are my cariables), 2^b = a and then 2^b + b +c = 17
But then couldn't solve.

(N is for intersection set, and n is for number of elements )
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November 14th, 2018, 11:52 PM   #2
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$A \cap B \neq \emptyset$

$|A \cup B | = 17 = |A| + |B| - |A \cap B|$

$2^{|A \cap B|} = |A-B| = |A| - |A \cap B|$

$|A| - |A \cap B| = 17 - |B|$

$2^{|A \cap B|} = 17 - |B|$

$|B| = 17 - 2^{|A \cap B|}$

$|A \cap B|=4 \Rightarrow |B|=1$

But this is impossible as $|A \cap B| \leq |B|$

$|A \cap B| = 3 \Rightarrow |B|=9$

This is perfectly valid. Further reducing $|A\cap B|$ just increases $|B|$

and thus $|B|=9$ is the minimum value.
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November 15th, 2018, 01:34 AM   #3
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Quote:
Originally Posted by romsek View Post
$A \cap B \neq \emptyset$

$|A \cup B | = 17 = |A| + |B| - |A \cap B|$

$2^{|A \cap B|} = |A-B| = |A| - |A \cap B|$

$|A| - |A \cap B| = 17 - |B|$

$2^{|A \cap B|} = 17 - |B|$

$|B| = 17 - 2^{|A \cap B|}$

$|A \cap B|=4 \Rightarrow |B|=1$

But this is impossible as $|A \cap B| \leq |B|$

$|A \cap B| = 3 \Rightarrow |B|=9$

This is perfectly valid. Further reducing $|A\cap B|$ just increases $|B|$

and thus $|B|=9$ is the minimum value.
thanks and by the way how to write the symbols as you wrote here?
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November 15th, 2018, 02:19 AM   #4
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Originally Posted by ketanco View Post
thanks and by the way how to write the symbols as you wrote here?
the board supports LaTex

a quick cheat sheet is here
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November 15th, 2018, 04:15 AM   #5
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Quote:
Originally Posted by romsek View Post
the board supports LaTex

a quick cheat sheet is here
so let me test: to write alpha, i must write slash and then the word?
\alpha

well i couldnt do it
i mean how do you write those symbols in here...
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November 15th, 2018, 04:47 AM   #6
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Originally Posted by ketanco View Post
so let me test: to write alpha, i must write slash and then the word?
\alpha

well i couldnt do it
i mean how do you write those symbols in here...
You have to type it inside "math" tags (or equivalently \$ ).

[math]\alpha[/math] gives $\displaystyle \alpha$

-Dan
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