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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