Set Theory Proof : Inclusion-Exclusion Principle
Not what you're looking for?
3. This exercise is about the inclusion-exclusion principle.
a) Let X and Y be finite ts and suppose that |X| = 11, |Y| = 6, and
|X∩Y| =4. Find |XUY|.
b) Suppose that U is a finite universal set. If |U| = 21, |XUY| = 11. |X| = 4 and |Y|= 10. find |XcUYc|.
c) Each tile in a collection of 19 is a square or a triangle and is also red or blue. Suppose that 12 of the 19 tiles are squares. 11 are red, and 4 are blue squares. Using the inclusion-exclusion principle, determine:
(1) the number of tile which are square or blue;
(2) the number of tiles which are triangles and red;
(3) the number of tiles which are red or squares.
[Hint: Let U be the (universal) set of all the 19 tiles and let S. T. R, and B denote the subset of tiles which is a square (respectively a triangle, red, and blue). Then T = SC and R = B. Part (2) comes down to computing |T∩R| and part (3) to determining |BcUTc|.
Purchase this Solution
Solution Summary
The Inclusion-Exclusion Principle is investigated.
Solution Preview
Please see the attached file for the complete solution.
Thanks for using BrainMass.
Solution.
a)
b) Since , we have
We know that
Note that |U|=21, we have
...
Education
- BSc , Wuhan Univ. China
- MA, Shandong Univ.
Recent Feedback
- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
- "excellent work"
- "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
- "Thank you"
- "Thank you very much for your valuable time and assistance!"
Purchase this Solution
Free BrainMass Quizzes
Solving quadratic inequalities
This quiz test you on how well you are familiar with solving quadratic inequalities.
Multiplying Complex Numbers
This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.
Probability Quiz
Some questions on probability
Know Your Linear Equations
Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.
Geometry - Real Life Application Problems
Understanding of how geometry applies to in real-world contexts