# Problem Involving Counting of Elements in Three Sets

Problem:
-The Union of set A, set B and set C has 104 elements.
-The Union of Set A and B has 51 elements
-The Union of Set A and C has 84 elements
-The Union of Set B and C has 97 elements
-The Intersection of Set A and the Union of Set B and C has 17 elements.
-Set C has twice as many elements as set B and three times as many elements as Set A.

How many elements does A have?

Hint: Take C is 6x and solve for x.

This is what I did after drawing a Venn diagram:

A minus B$\cup$C = 104-97=7
B minus A$\cup$C = 104-84=20
C minus A$\cup$B = 104-51=53

If you add up these numbers and subtract them from 104, you'll get that all the intersections of these sets together have 24 elements.

It is given that A$\cap$(B$\cup$C) = 17.

Therefore, the intersection of B and C without whatever is in A, should be 7.

From this fact the elements of A can be calculated from the fact that A$\cup$C has 84 elements. To calculate the elements of A take 84 and subtract the elements of C that are not in A, 84 -53 -7= 24

I didn't use the hint nor did I use the size relationship between the sets, so I am not sure if I did this problem right.

How could I solve this problem with the hint (solve as an equation of x) and the given relationship between A, B and C?

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-Set C has twice as many elements as set B and set B has twice as many elements as Set C.
Huh?

Sorry, typo.

Set C is twice as large as B and three times as large as A.

Problem:
-The Union of set A, set B and set C has 104 elements.
-The Union of Set A and B has 51 elements
-The Union of Set A and C has 84 elements
-The Union of Set B and C has 97 elements
-The Intersection of Set A and the Union of Set B and C has 17 elements.
-Set C has twice as many elements as set B, and three times as many elements as Set A.

How many elements does A have?
You only need to use three of the above 7 conditions to determine the size of A, so hopefully there are some interesting follow-up questions to your problem. Given that unhelpful hint, that does not seem too likely.

The hint and the size relationship confused me. I don't see how to solve this problem as an equation of x and given the fact that C is twice as large as B and three times as A, but I guess it is possible. If someone sees it, I am curious how to proceed.

There is no follow up question to this problem.

The teacher of this course is a little bit nuts though, on another test he asked to prove that every even integer greater than 4 can be written as the sum of two primes for extra credit.

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