The Principle of Inclusion-Exclusion

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Presentation transcript:

The Principle of Inclusion-Exclusion Example 1: In a discrete mathematics class every student is a major in computer science or mathematics, or both. The number of students having computer science as a major (possibly along with mathematics) is 25; the number of students having mathematics as a major (possibly along with computer science) is 13; and the number of students majoring in both computer science and mathematics is 8. How many students are in this class? FIGURE 1 The Set of Students in a Discrete Mathematics Class.

The Principle of Inclusion-Exclusion Example 2: How many positive integers not exceeding 1000 are divisible by 7 or 11? HW: Example 3,(p.501) FIGURE 2 The Set of Positive Integers Not Exceeding 1000 Divisible by Either 7 or 11.

The Principle of Inclusion-Exclusion |A∪B∪C|= |A|+|B|+|C|-|A ∩ B|-|A ∩ C|-|B ∩ C|+|A ∩ B ∩ C| FIGURE 3 Finding a Formula for the Number of Elements in the Union of Three Sets.

The Principle of Inclusion-Exclusion Example 4: A total of 1232 students have taken a course in Spanish, 879 have taken a course in French, and 114 have taken a course in Russian. 103 have taken courses in both Spanish and French, 23 have taken courses in both Spanish and Russian , and 14 have taken course in both French and Russian. If 2092 students have taken at least one of Spanish, French, and Russian. how many students have taken a course in all three languages? FIGURE 4 The Set of Students Who Have Taken Courses in Spanish, French, and Russian.

The Principle of Inclusion-Exclusion Theorem 1: The Principle of Inclusion-Exclusion Let A1, A2,. . ., An be finite sets. Then Example 5: Give a formula for the number of elements in the union of four sets.