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If |X| is the number of elements in set X, and “∪� is

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by Max@Math Revolution » Tue Oct 16, 2018 11:28 pm

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[Math Revolution GMAT math practice question]

If |X| is the number of elements in set X, and "∪" is the union and "∩" is the intersection of 2 sets, what is the value of |A∩B|?

1) |A∪B|�50
2) |B|=40
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Source: — Data Sufficiency |
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by rrnicky » Wed Oct 17, 2018 4:27 am
IAUBI = IAI + IBI - IAnBI
IAUBI and IBI known but IAI not known so information is not sufficient
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by fskilnik@GMATH » Wed Oct 17, 2018 7:54 am
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

If |X| is the number of elements in set X, and "∪" is the union and "∩" is the intersection of 2 sets, what is the value of |A∩B|?

1) |A∪B|�50
2) |B|=40
$$?\,\,\,:\,\,\,\# \,\,\,{\rm{common}}\,\,{\rm{elements}}$$
Let´s go straight to (1+2) to show a BIFURCATION:
$$\left\{ \matrix{
\,\,{\rm{Take}}\,\,B = \left\{ {1,2,3, \ldots ,38,39,40} \right\}\, \hfill \cr
\,\,{\rm{Take}}\,\,A = \left\{ {41,42,43, \ldots ,49,50} \right\} \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 0$$
$$\left\{ \matrix{
\,\,{\rm{Take}}\,\,B = \left\{ {1,2,3, \ldots ,38,39,40} \right\}\, \hfill \cr
\,\,{\rm{Take}}\,\,A = \left\{ {1,2,3, \ldots ,48,49,50} \right\} \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 40$$

The correct answer is therefore (E).

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
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by Max@Math Revolution » Thu Oct 18, 2018 11:54 pm
=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.
Note that
|A∪B| = |A| + |B| - |A∩B| and |A∩B| = |A| + |B| - |A∪B|.

Since we have 4 variables (|A∩B|, |A|, |B|, |A∪B|) and 1 equation (|A∩B| = |A| + |B| - |A∪B|), E is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
Suppose A and B are disjoint sets, |A∪B| = 50, |A| = 10, and |B| = 40. Then |A∩B| = |A| + |B| - |A∪B| = 0.
Suppose A contains B, |A∪B| = 50, |A| = 50, and |B| = 40. Then |A∩B| = |A| + |B| - |A∪B| = 40.
Since we don't have a unique solution, both conditions together are not sufficient.


Therefore, E is the answer.
Answer: E
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