Math Is Fun Forum

Toast

Real Member

Registered: 2006-10-08

Posts: 1,321

Principle of Inclusion and Exclusion

There's this section in my textbook about the principle of inclusion and exclusion, and they have this formula:

If n(A) = set A
and n(B) = set B,
n(A or B) = n(A) + n(B)  - n(A and B)

Should 'n(A or B)' really be 'n(A and/or B)'?
Wouldn't 'n(A or B)' = n(A) + n(B) - 2n(A and B)?

This diagram was there to help explain too.

Last edited by Toast (2007-01-03 14:19:45)

Ricky

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Registered: 2005-12-04

Posts: 3,791

Re: Principle of Inclusion and Exclusion

What is n(A)?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Toast

Real Member

Registered: 2006-10-08

Posts: 1,321

Re: Principle of Inclusion and Exclusion

The number of objects which satisfies condition A.

Toast

Real Member

Registered: 2006-10-08

Posts: 1,321

Re: Principle of Inclusion and Exclusion

I'll also give the specific example that prompted my question.

ExampleIn a class of 26 students, 18 like English and 12 like Science. Assuming each student likes at least one of English and Science, how many student like both?

Solution
Let n(E) = number of students who like English
and n(s) = number of students who like Science
Then by principle of inclusion and exclusion, we have
         n(E or S) = n(E) + n(S) - n(E and S)
i.e.              26 = 18 + 12 - n(E and S)
so               26 = 30 - n(E and S)
∴    n(E and S) = 4
i.e. four students like both English and Science

I don't think n(E or S) should be representative of the class, because it really discludes n(E and S) altogether (the shared section of the Venn Diagram).
n(E and/or S) is representative of the whole venn diagram.

Right?

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Principle of Inclusion and Exclusion

I think your getting inclusion and exclusion or mixed up.  Inclusive or is where it can be in either A or B, or both.  Exclusive or is where it can be in A or B, but not both.

There is no such thing as and/or (it's simply not called that, although thats the general idea), and typically, unless stated otherwise, all or's are inclusive.

So:

I don't think n(E or S) should be representative of the class, because it really discludes n(E and S)

Does in fact include n(E and S).

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Principle of Inclusion and Exclusion

As Ricky says, if you see an 'or' somewhere, it generally means and/or. The or that doesn't include and is represented by xor.

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Why did the vector cross the road?
It wanted to be normal.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Principle of Inclusion and Exclusion

But there are some pitfalls here, so it is important to be careful.

For example, "Ben likes basketball or baseball" typically is interpreted as being exclusive or, that is, xor.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Toast

Real Member

Registered: 2006-10-08

Posts: 1,321

Re: Principle of Inclusion and Exclusion

Ok, so if there are 24 students in a class, 16 of which play tennis and 14 of which play soccer, would I express those who play only one sport as:

n(T xor S) = n(only T) + n(only S)
OR
n(T xor S) = n(T or S) - n(T and S)

n(T xor S) = 10 + 8
OR
n(T xor S) = 24 - 6

∴ n(T xor S) = 18

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Principle of Inclusion and Exclusion

Correct.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Toast

Real Member

Registered: 2006-10-08

Posts: 1,321

Re: Principle of Inclusion and Exclusion

Sorry, I'd just like to make sure that n(only T) is the right notation for people who only play tennis? Is it right?

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Principle of Inclusion and Exclusion

Technically, yes.  However, typically in problems you aren't given that.  You are just given n(T) and n(S).

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Ricky

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Registered: 2005-12-04

Posts: 3,791

Re: Principle of Inclusion and Exclusion

A better way to say it is:

n(T \ S) which is the set T excluding all elements of T which are also in S.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Toast

Real Member

Registered: 2006-10-08

Posts: 1,321

Re: Principle of Inclusion and Exclusion

Thank everyone, I think I'm all set for this chapter.

Ricky

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Registered: 2005-12-04

Posts: 3,791

Re: Principle of Inclusion and Exclusion

That's good to hear.  Do you have a test on it soon, or is this just a subset of all the things you need for your next exam?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Toast

Real Member

Registered: 2006-10-08

Posts: 1,321

Re: Principle of Inclusion and Exclusion

Well, yes, a subset of everything needed for my next exam. You see, I've been consistently working through my textbook for next year, doing every problem. With a lot more work and some luck I'll be able to complete it before the start of the school year.