The diagrams drawn to represent sets are called Venn diagrams or Euler -Venn diagrams. Here we represent the universal set U by points within rectangle and the subset A of the set U represented by the interior of a circle. If a set A is a subset of a set B then the circle 1 representing A is drawn inside the circle representing B. If A and B are no equal but they have some common elements, then to represent A and B by two intersecting circles.

Illustration -: A class has 175 students. The following table shows the number of students studying one or more of the following subjects in this case

                        Subjects                                             No. of students

                        Mathematics                                                          100

                        Physics                                                                         70

                        Chemistry                                                                  46

                        Mathematics and Physics                               30

                        Mathematics and Chemistry                        28

                        Physics and Chemistry                                      23

                        Mathematics, Physics and Chemistry  18

                        How many students are enrolled in Mathematics alone, Physics alone and Chemistry alone? Are there students who have not offered any one of these subjects?

Solution: Let P, C, M denotes the sets of students studying Physics, Chemistry and Mathematics respectively.

Let a, b, c, d, e, f, g denote the number of elements ≤ students) contained in the bounded region as shown in the diagram then

a + d + e + g = 70
c + d + f + g = 100
b + e + f + g = 46
d + g = 30
e + g = 23
f + g = 28
g = 18

after solving we get g = 18, f = 10, e = 5, d = 12, a = 35, b = 13 and c = 60
Therefore a + b + c + d + e + f + g = 153
So, the number of students who have not offered any of these three subjects = 175 –153 = 22
Number of students studying Mathematics only, c = 60
Number of students studying Physics only, a = 35
Number of students studying Chemistry only, b = 13.

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