# Math Help - refreshment on sets

1. ## refreshment on sets

In an examination , part A was attempted by 60 students , part B by 52 students and part C by 45 students . 22 students attempted both parts A and B , 12 attempted both parts B and C , 20 attempted parts A and C and 4 attempted all three parts .

(a) How many students attempted part A but not parts B and C ?

(b) How many students attempted part Bbut not parts A and C ?

(c) How many students attempted at least 2 parts ?

For (A) , n(A-(B n C))=60-12=48 but it is wrong .

i tried the same method on B and i hv no idea at all for c .

THanks .

2. I presume you know the answers (as you know that one is wrong) so I'll give you what I got:
1) 22
2) 22
3) 46

How I got them? Well, I started by looking at the unions of the sets:
$|A \cup B| = |A|+|B|-|A \cap B|$, etc. Then, I couldn't think up some clever way so I decided to do the obvious thing and...draw a diagram!

Draw three circles which all intersect one another. Label them A, B and C. There should be a triangle in the middle where they all intersect one another - this represents the set $A \cap B \cap C$. Similarly, where the circles A and B intersect this represents the intersection of the sets A and B, etc. We shall now put numbers in the spaces to represent the size of the sets.

So you know what to put in $(A \cap B) \cap C$ - that's a 4. What about in $(A \cap B) \setminus C$ - that's simply 22-4=18. Then just continue putting numbers into this diagram until every space has a number in it. Questions 1 and 2 are easily read from this diagram, and question 3 isn't really hard to see on the diagram either...

I hope that makes sense, and also that it helps!

EDIT: essentially, the answer to part 1 is the cardinality of the set $(A \setminus B) \setminus C$, which is $|A| - |A \cap C| - |A \cap B| + |A \cap B \cap C|$ as you have "removed" $A \cap B \cap C$ twice (as it is in both $A \cap C$ and $A \cap B$) so you need to put one of them back.

EDIT 2: Your diagram should look something like
http://www.biomedcentral.com/content...-9-444-3-l.jpg
just with different numbers, and less stolen from someone else's web site...(it's too big to put in my post properly)