Fuzzy(2)
- 1. TOPIC: venn diagram and sets theory PRESENTED BY:- HIMANSHU GAWANDE (A-55) DEPARTMENT OF ELECTRONICS ENGINEERING (3RD YR/6TH SEM) GUIDED BY:- PROF .Sayankar
- 2. Venn Diagrams • In set theory, the universe of discourse is called the universal set, typically designated with the letter U. Venn Diagrams were developed by the logician John Venn (1834 – 1923). In these diagrams, the universal set is represented by a rectangle and other sets of interest within the universal set are depicted as circular regions.
- 3. Venn Diagrams • The rectangle represents the universal set, U, while the portion bounded by the circle represents set A. A U
- 4. Complement of a Set • For any set A within the universal set U, the complement of A, written A', is the set of all elements of U that are not elements of A. That is { | and }.A x x U x A
- 5. Subsets of a Set • Set A is a subset of set B if every element of A is also an element of B. In symbols this is written .A B B AU
- 6. • 1. From a survey of 100 college students, a marketing research company found that 75 students owned stereos, 45 owned cars, and 35 owned cars and stereos. • a) How many students owned either a car or a stereo? • b) How many students did not own either a car or a stereo?
- 7. • METHOD: • a) Start with a Venn Diagram and label the different categories: b) Fill in the number of students who own both cars and stereos, which would be in the intersection of the two sets:
- 8. • c) Fill in the remaining numbers for the two sets. • In this case, since a total of 45 students own cars, and 35 have already been listed, then 45 - 35 = 10 students own cars only. • Similarly, since 75 students own stereos and 35 have already been listed, then 75 - 35 = 40 students who own stereos only: d) Finally, interpret and answer the questions:
- 9. • How many students owned either a car or a stereo? The question asks either ... or which is union of the sets. From the diagram, the number of elements in • A = 10 + 35 and the number of elements in B which are NOT in A are 40. So the union would be 10+35+40 = 85 How many students did not own either a car or a stereo? The question asks for the number not in either A nor B (namely, the complement of A B or (A B)' ). Since there are 100 students in the universe, then the complement is found by subtracting those who own either a car or stereo from the total number of students surveyed or 100 - 85 = 15.