Inclusion-Exclusion Principle

Date: 09/03/2002 at 19:56:57
From: Rehana Chandarpal
Subject: Sets

In a survey of 100 people, 85 like calypso and 93 like pan. Calculate 
the number of people who like both calyso and pan.

I tried answering this question by doing the equation that my teacher 
showed me but I still can't do it.

Date: 10/18/2002 at 17:27:22
From: Doctor Nitrogen
Subject: Re: Sets

Hi, Rehana:

Why don't we let a capital letter like X denote a set? And we'll use 
#X to denote the number of elements in set X. 

Let U be the set with 100 people in it, let A be the subset of U that 
has all the people who like calypso, let B be the subset of people in 
U who like pan, and let C be the subset of people in U who like both 
calypso and pan. Then

                 #U = 100.

                 #A = 85

                 #B = 93

                 #C = (unknown).

To find #C, use the fact that the total number of people in U equals 
the total number of people in A plus the total number of people in B, 
minus the total number of people in C, so

                 #U = #A + #B - #C
or
                 100 = 85 + 93 - #C

Can you figure the rest out now? This way of solving a math problem 
like this one uses what is called the "Inclusion-Exclusion Principle." 
Here it uses the fact that

                 #U = #(A u B) - #(A intersection B).

The lower case "u" denotes "union."

Did this help answer the question you had concerning your mathematics 
problem? You are welcome to return to The Math Forum/Doctor Math 
whenever you have any math related questions.

- Doctor Nitrogen, The Math Forum
  http://mathforum.org/dr.math/