MHB How Many Students Are Not Involved in Any Afternoon Activities?

[Yankel]

Hello all,

I am struggling with this relatively simple task.

In a university with 88 students, each student can choose to participate in 3 afternoon activities: activity A, activity B and activity C. Each student can choose to participate in some activities, all or none.

33 students participate in activity A
28 students participate in activity B
33 students participate in activity C
14 students participate in activity A and B
18 students participate in activity A and C
10 students participate in activity B and C
6 students participate in activity A, B and C

1. How many students decided not to participate in any activity ?
2. How many students participate ONLY in activity A ?
3. How many students participate in activity A OR B, but NOT in C ?

I think I did "1" OK, I got that the answer is 30 (am I correct ?).

I solved it using union and intersection, and using the rule of union of 3 sets.

I find it hard to solve "2" and "3".

 

[MarkFL]

You are correct for question 1. I find a Venn diagram is an invaluable tool for problems of this sort. Draw 3 intersecting circles to represent the 3 activities, and then work from the inside out, that is, start with the intersection of all 3, then fill in the 3 intersections of the pairs, and finally the 3 parts of each set with no intersection. You will then have all the information you need to easily answer the remaining 2 questions.

Can you proceed?

 

[Yankel]

I think I understand what you mean by inside out, I think I can proceed, thank you !

Just for curiosity, isn't it also possible to be done using algebra of sets ?

 

[MarkFL]

...
Just for curiosity, isn't it also possible to be done using algebra of sets ?

My inclination is that it is, but this is not an area in which I am very knowledgeable. I just find a Venn diagram to be very straightforward for a problem like this. :D

 

[Yankel]

Ok, thanks :)

Just to verify, in "2" and "3", are the answers 7 and 10 ?

 

[MarkFL]

I agree with 7 for question 2, but for question 3 I have a different answer. I get 10 as the number of students participating only in activity B, but we need to include those that participate only in A AND in A and B (but not C).