5_Inclusion-Exclusion_More_Counting

# 5_Inclusion-Exclusion_More_Counting - Discrete Mathematics...

This preview shows pages 1–9. Sign up to view the full content.

Discrete Mathematics Dr. Fred Phelps Lecture 5 The Inclusion- Exclusion Principle & More Counting Problems

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
LPV 2.3 & Chen 13 The Inclusion-Exclusion Principle This is covered in LPV 2.3 but the lecture is taken from Chen’s book. Consider the sets S = {1, 2, 3, 4}; T = {1, 3, 5, 6, 7} and W = {1, 4, 6, 8, 9}. Suppose that we would like to count the number of elements of their union We might do this in the following way: 1. We add up the numbers of elements of S, T and W. Then we have the count . S T W U U 4 5 5 14. S T W + + = + + =
Chen 13 - The Inclusion-Exclusion Principle S = {1, 2, 3, 4}; T = {1, 3, 5, 6, 7} and W = {1, 4, 6, 8, 9}. Find 1. Note that 2. Clearly we have over-counted. For example, the number 3 belongs to S as well as T , so we have counted it twice instead of once. 3. We compensate by subtracting the number of those elements which belong to more than one of the three sets S , T and W . . S T W U U 4 5 5 14. S T W + + = + + = 14 2 2 2 8. S T W S T S W T W + + - - - = - - - = I I I

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Chen 13 The Inclusion-Exclusion Principle S = {1, 2, 3, 4}; T = {1, 3, 5, 6, 7} and W = {1, 4, 6, 8, 9}. Find Note that But now we have under-counted. For example, the number 1 belongs to all the three sets S , T and W , so we have counted it 3-3 = 0 times instead of once. We therefore compensate again by adding the number of those elements which belong to all the three sets S , T and W . Then we have the count . S T W U U 8. S T W S T S W T W + + - - - = I I I 8 1 9 S T W S T S W T W S T V + + - - - + = + = I I I UU S T V U U
The Inclusion-Exclusion Principle A B C A B C A B A C B C A B C = + + - - - + I I I I I U U

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
The Inclusion-Exclusion Principle A B C A B C A B A C B C A B C = + + - - - + I I I I I U U
The Inclusion-Exclusion Principle

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
The Inclusion-Exclusion Principle Provin g
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern