# Thread: Cardinalities and ultimate sets

1. ## Cardinalities and ultimate sets

When the prof was lecturing on cardinality of sets, and ultimate sets, everything seemed quite straightforward. So when I saw this question I thought "No problem"

To my dismay I am completely vexed by it.

In a sample of 1000 cottage owners, 570 owned their cottages with no mortgage and also owned cars; 340 owned mortgaged cottages, owned cars, and had electricity; 189 had neither mortgages nor cars nor electricity; 760 owned cars. Find the cardinalities of the ultimate sets.

A=mortage
B=Car
C=electricity

I am just going to represent not with a ! sign and all sets are intersect.

So the ultimate sets are..

!A !B !C=
A !B !C=
!A B !C=
!A !B C=
A B !C=
A !B C=
!A B C=
A B C=

"570 owned their cottages with no mortgage and also owned cars"

To me this means no mortgage and car which is !A C=570

" 340 owned mortgaged cottages, owned cars, and had electricity"

this is the intersection of |ABC| which is an ultimate set.

"189 had neither mortgages nor cars nor electricity;"

this is an ultimate set |!A!B!C|=189

"760 owned cars"

This is simply |B|=760

So we have

|B|=760
|!A!B!C|=189
|ABC|=340
|U|=1000
|!AC|=570

First does this seem like the correct interpretation? Now I just keep trying to make conclusions from this information, but keep getting lost.

2. ## Re: Cardinalities and ultimate sets

a hint:

1000 cottage owners = universal (or ultimate set, depending on what book you're using)

570 owned their cottages with no mortgage and also owned cars = |$\displaystyle A^c$$\displaystyle B| 340 owned mortgaged cottages, owned cars, and had electricity = |\displaystyle ABC| 189 had neither mortgages nor cars nor electricity = |\displaystyle A^c$$\displaystyle B^c$$\displaystyle C^c$|

760 owned cars = |$\displaystyle B$|

That should get you started.

P.S. ehpoc, did you get my PM?

3. ## Re: Cardinalities and ultimate sets

If we are to be helpful, we must be able to read the notation.
You seem to have the misfortune to be saddled with completely non-standard notation.

What does $\displaystyle !A$ mean? Ultimate ??
It appears this is an ordinary Venn Diagram problem.

4. ## Re: Cardinalities and ultimate sets

I noted in my thread that !=not aka !A=everything outside of A.

Regular notation it is noted with a bar over it. A bar.

1000 cottage owners = universal (or ultimate set, depending on what book you're using)

570 owned their cottages with no mortgage and also owned cars = ||

340 owned mortgaged cottages, owned cars, and had electricity = ||

189 had neither mortgages nor cars nor electricity = ||

760 owned cars = ||
This is essentially what I cam up with, but I am having trouble deducing anything further about the ultimate sets.

Like I can come to some clear conclusions, but I can not get close to knowing the cardinality of each individual set.

Like ok

|B|=760
|!A!B!C|=189
|ABC|=340
|U|=1000
|!AC|=570

I know that |AuBuC|=811

and I can then conclude that |A u B^c u C| is 51

EDIT: Ok now I am really confused. Look at the following.

570 owned their cottages with no mortgage and also owned cars = ||

This says 570 are in the area of B that is not in A...correct?

340 owned mortgaged cottages, owned cars, and had electricity = ||

This says that there is 340 in the area of B that is shared with A

570 in the area of B that is not in A
At least 340 in the area of B that is shared with A
760 in B in total

Ummm is there a problem here?

5. ## Re: Cardinalities and ultimate sets

This says 570 are in the area of B that is not in A...correct?
Correct. Another way: in the set of $\displaystyle A^c$ $\displaystyle \cap$ $\displaystyle B$

This says that there is 340 in the area of B that is shared with A
$\displaystyle A \cap B \cap C$

In all those areas (in A AND B AND C)

6. ## Re: Cardinalities and ultimate sets

You should draw a Venn Diagram.
I have a beautiful venn drawn on my whiteboard right now.

Do you see the contradiction though?

These three premises have been stated by the question....

570 in the area of B that is not in A
At least 340 in the area of B that is shared with A
760 in B in total

That is not possible

7. ## Re: Cardinalities and ultimate sets

You are correct. As stated, the problem is not possible, because there would have to be negative 150 people with mortgages and cars but no electricity. Either the problem is wrong, or you typed it in wrong.

8. ## Re: Cardinalities and ultimate sets

Here is the exact problem copy and pasted from the assignment. Can yo confirm I am not crazy and that there is indeed a typo?

3. In a sample of 1000 cottage owners, 570 owned their cottages with no mortgage and also owned cars; 340 owned mortgaged cottages, owned cars, and had electricity; 189 had neither mortgages nor cars nor electricity; 760 owned cars. Find the cardinalities of the ultimate sets.

9. ## Re: Cardinalities and ultimate sets

If that's the entire problem and there is no other information given in relation to the problem, then there is a typo.

10. ## Re: Cardinalities and ultimate sets

Is the problem these 3 premesis?

-570 in the area of B that is not in A
-At least 340 in the area of B that is shared with A
-760 in B in total

11. ## Re: Cardinalities and ultimate sets

Originally Posted by ehpoc
Is the problem these 3 premesis?
Yes. Simple math tells you that 910 > 760, so the only way to fit in the extra people is to cancel them out with negative people, which (unless you're looking at a time series change or something, perhaps that's the ticket) is impossible.

12. ## Re: Cardinalities and ultimate sets

Apparently the question intentionally uses improper data and the cardinality of the ultimate sets is a way to detect it....

So I am guess the set of A intersect B is going to be negative or something.

I still have no clue how I can find the cardinality of all of the ultimate sets from the information given.

13. ## Re: Cardinalities and ultimate sets

If negative values are permitted, then it is not remotely possible to determine the number in each section of the Venn diagram with the information you have. For a solution to be possible, you would need at least 8 numbers given to you by the problem (with the correct overlaps), and you don't have 8 numbers, so it's impossible.

Also, please don't say "cardinality of set" if you're talking about negative numbers. Sets can't hold anti-items. If these are sets, then they do not contain "negative people". The cardinality of a set is always an ordinal number (for finite sets this is a natural number).

14. ## Re: Cardinalities and ultimate sets

Well then I have no clue what is going on. Everyone is telling me this problem is impossible then?!?!?

15. ## Re: Cardinalities and ultimate sets

Originally Posted by ehpoc
Well then I have no clue what is going on. Everyone is telling me this problem is impossible then?!?!?
Yes. It's impossible because it gives you equations that produce a negative result for one of the sets. Even if sets could hold "fewer than zero" elements, which they cannot, if negative numbers are allowed in the Venn then determining all of the values would require more data than you're given.

It is probably just a typo in the problem. There aren't supposed to be negative people, and you would have enough information to determine what you need if the numbers were right. Email your instructor already.

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