Venn Diagram Problem

[Jalenke]

Hi. Sorry if I'm posting this in the wrong area, I wasn't sure where to post. It's Liberal Arts math. The question is: A universal set U consists of 21 elements. If sets A, B, and C are proper subsets of U and n(U)=21, n(A∩B)=n(A∩C)=n(B∩C)=8, n(A∩B∩C)=3, and n(AUBUC)=18,Determine
a) n(AUB)
b) n(A'UC)
c) n(A∩B)'
If I could figure out how to fill out the diagram I know how to answer the problem. So I think that 3 would go in the middle where all circles overlap and 5 would go in the B and C overlap. I’m really lost with the rest. Can anyone help me? Thanks!

 

[tkhunny]

Hi. Sorry if I'm posting this in the wrong area, I wasn't sure where to post. It's Liberal Arts math. The question is: A universal set U consists of 21 elements. If sets A, B, and C are proper subsets of U and , n(A∩B)=n(A∩C)=n(B∩C)=8, ,Determine
a) n(AUB)
b) n(A'UC)
c) n(A∩B)'
If I could figure out how to fill out the diagram I know how to answer the problem. So I think that 3 would go in the middle where all circles overlap and 5 would go in the B and C overlap. I’m really lost with the rest. Can anyone help me? Thanks!

Do you know what a "Proper" subset is?
n(U)=21 The entire Universe is 21. A proper subset cannot be any greater than 20.
n(AUBUC)=18 The union of all three of A, B, and C is 18. This leaves 21-18 = 3 out in the cold.
n(A∩B∩C)=3 Yes, this makes the very center, where all three circles meet, '3'.
n(A∩B)=n(A∩C)=n(B∩C)=8 This makes the flower petal regions each 8. There should be three such sections. However, the very center is already included in this count. You must discard it This makes each flower petal section a combination of 5 and 3, making 8 total.

Not many boxes left to fill!

 

[mmm4444bot]

n(A∩B) = n(A∩C) = n(B∩C) = 8


c) n(A∩B)

Please check your typing. It appears (from the given information above) that they provided you with the answer to (c).

I think that 3 would go in the middle where all circles overlap and 5 would go in the B and C overlap.

That's a reasonable first start. The given information states that the B-C overlap equals the A-B and A-C overlaps, too. (They are all 8.)


Remember that not all of the 21 elements need be inside the circles. That is, some elements in the universal set may lie outside the circles.

 

[JeffM]

Hi. Sorry if I'm posting this in the wrong area, I wasn't sure where to post. It's Liberal Arts math. The question is: A universal set U consists of 21 elements. If sets A, B, and C are proper subsets of U and n(U)=21, n(A∩B)=n(A∩C)=n(B∩C)=8, n(A∩B∩C)=3, and n(AUBUC)=18,Determine
a) n(AUB)
b) n(A'UC)
c) n(A∩B)'
If I could figure out how to fill out the diagram I know how to answer the problem. So I think that 3 would go in the middle where all circles overlap and 5 would go in the B and C overlap. I’m really lost with the rest. Can anyone help me? Thanks!

You appear to have edited what you originally posted as question c. Please do not do that. Make corrections in a separate post within the same thread. Otherwise, the thread becomes very confusing.

With respect to question c

What is \(\displaystyle n\left(P \bigcup Q\right)?\)

This should be in your book or class notes, but make up a few sets to convince yourself of why this is correct.

Let U be the universal set. Let K be a subset of U.

What is \(\displaystyle K \bigcup K'\)? If you translate the symbols into English, you will see the answer.

So what is \(\displaystyle n\left(K \bigcup K'\right)\)? Give the answer in two different formats if you can.

What is \(\displaystyle K \bigcap K'\)? If you translate the symbols into English, you will see the answer.

So what is \(\displaystyle n\left(K \bigcap K'\right)\)?

So what is \(\displaystyle n(K')\)?

 

[pka]

Hi. Sorry if I'm posting this in the wrong area, I wasn't sure where to post. It's Liberal Arts math. The question is: A universal set U consists of 21 elements. If sets A, B, and C are proper subsets of U and n(U)=21, n(A∩B)=n(A∩C)=n(B∩C)=8, n(A∩B∩C)=3, and n(AUBUC)=18,Determine
a) n(AUB)
b) n(A'UC)
c) n(A∩B)'
If I could figure out how to fill out the diagram I know how to answer the problem. So I think that 3 would go in the middle where all circles overlap and 5 would go in the B and C overlap. I’m really lost with the rest. Can anyone help me? Thanks!

 

[Jalenke]

Hi everyone thanks for the responses. I wasn't aware I edited question C. I originally typed my question in Word and pasted it here. I noticed that the words weren't spacing out so I edited that part shortly after I posted. So I do get how you fill out all the middle parts. I still don't understand how to answer the questions. I'm sure you just explained it but I'm really bad at math and I don't see it. I already have the answer to this question I just don't understand hot to get them.

 

[JeffM]

Hi everyone thanks for the responses. I wasn't aware I edited question C. I originally typed my question in Word and pasted it here. I noticed that the words weren't spacing out so I edited that part shortly after I posted. So I do get how you fill out all the middle parts. I still don't understand how to answer the questions. I'm sure you just explained it but I'm really bad at math and I don't see it. I already have the answer to this question I just don't understand hot to get them.

Please answer, to the best of your ability, to answer the questions in my previous post. They represent an effort to take you step by step through the logic

Edit By the way, I majored in European languages and history and am highly supportive of the idea that those studying the liberal arts can and should be trained in mathematics.

Further Edit: If you check mmm's post, you will see that the question c that he copied and the question c in your edited original post differ. That indicates to me that you changed your original post while mmm was working on his post. Just to make sure, is question c what is in the edited version of your original post or what is in the version copied by mmm? They are quite different questions.