Using Set Theory

1. Using Set Theory • A ~ Set of Students Studying Math • B ~ Set of Students Studying History • A∩B ~ The intersection of sets A and B. • (Students who study Math and History) S

2. 2 Sets are disjoint if their intersection is the empty set Ø • Disjoint Sets e.g. if C is the set of students studying grade 10 history and A is the set of students studying grade 12 math there are no grade 10 history students who study grade 12 math.

3. A U B ~ The union of sets A and B, The set of all the elements in either A or B or both

4. n(A) = the number of elements in set A • A’ = the set of elements not in A (called A’s compliment)

5. Union of Sets • A U B = {set of element in A or B (or both)} • If n(A UB) = n(A) + n(B) the sets are disjoint.

6. The Union of Sets Generally • n( A UB) = n( A) + n( B) – n( A∩B ) • The number of members of set A or B

7. n( A UB) = n( A) + n( B) – n( A∩B ) • e.g. A = set of DM Students B = set of P.E. Students There are 28 DM students and 25 PE students. 12 are in both. How many students in total are in DM or PE (or both)? n(A)=28, n(B)=25, n( A∩B )=12 n( A UB) = n( A) + n( B) – n( A∩B ) =28 + 25 – 12 = 41

8. Example 2 Of the 120 students in a class, 30 speak Chinese, 50 speak Spanish, 75 speak French, 12 speak Spanish and Chinese, 30 speak Spanish and French, and 15 speak Chinese and French. Seven students speak all three languages. How many students speaks none of these languages?

9. 7 speak all 3 15 speak Chinese and French 15 30 speak Spanish and French 12 speak Spanish and Chinese 23 5 50 speak Spanish 7 10 30 speak Chinese 37 8 75 Speak French 120 – 105 = 15 don’t speak any.

10. Homework • Page 270 • 1 to 8