A Fundamental Principle of Counting

1. A Fundamental Principle of Counting

2. The Importance Counting problems arise in many applications of mathematics and comprise the mathematical field of combinatorics. We’ll study a number of different sorts of counting problems in the remainder of this chapter.

3. Cardinality of a Set The cardinality of a set refers to how many elements are in the set. If S is any set, we will denote the number of elements in S by n(S).

4. Example Let S = {2, 4, 6, 8, 10}. Find n(S). Let S = Ø. Find n(S).

5. A Class Example How many of you have a dog? n(D) = How many of you have a cat? n(C) = How many of you have both? n(D n C) = Using only the information above, determine the number of classmates who have a dog OR a cat. n(D n C) =

6. Inclusion-Exclusion Principle Let S and T be sets. Then, n(S U T) = n(S) + n(T) – n(S n T) The Inclusion-Exclusion Principle is sometimes referred to as the Union Rule.

7. Examples Find n(S n T), given that n(S) = 4, n(T) = 12, and n(S U T) = 15. Find n(T), given that n(S) = 14, n(S n T) = 6, and n(S U T) = 14.

8. Example: Course Enrollments Suppose that all of the 1000 first-year students at a certain college are enrolled in a math or an English course. Suppose that 400 are taking both math and English and 600 are taking English. How many are taking a math course?

9. Venn Diagrams Sets can be visualized geometrically by drawings known as Venn diagrams.

10. Shading Venn Diagrams Shading different regions of the rectangle can illustrate a number of sets.

11. Examples For each of the following, draw a Venn diagram and shade the portion corresponding to the set. a.) S' n T' b.) S U (T' U S ) c.) R n (S U T )