TOPIC: venn diagram and sets theory

PRESENTED BY:-

HIMANSHU GAWANDE (A-55)

DEPARTMENT OF ELECTRONICS ENGINEERING

(3RD YR/6TH SEM)

GUIDED BY:- PROF .Sayankar

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.

Venn Diagrams

• The rectangle represents the universal set, U, while the portion bounded by the circle represents set A.

A

U

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

Subsets of a Set

• Set A is a subset of set B if every element of Ais also an element of B. In symbols this is written

.A B

B

AU

• 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?

• 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:

• 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 = 10students 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:

• 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 + 35and 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.