The Venn diagram – Two-Set Problem.

In order to help solve problem involving sets, we use a diagram known as the Venn diagram. In a Venn diagram, each set is represented by an oval or circle. That is

The universal set ‘ U’ is represented by a rectangle as shown fig.1 below. The Venn diagram gives a clear picture of information. 

If  U is the universal set and A and B are two intersecting subset of the universal set then Fig.1, Fig.2, Fig.3, Fig.4, Fig.5 and Fig.6 show how to illustrate or represent the sets A and B in a Venn diagram.

Ø  Fig.1 below is A ∩ B′, i.e., elements belonging to only A.  The shaded region represents A ∩ B′.

Ø  Fig.2 is A ∩ B, i.e. elements belonging to both A and B. The shaded region represents A ∩ B.

Ø  Fig.3 is A′ ∩ B, i.e. elements belonging to only B. The shaded region represents A′ ∩ B.

Ø  Fig.4 is A U B i.e. all element in A and B. The shaded regions represent A U B.

Ø  Fig.5 is (A U B)′ i.e. elements belonging to neither A nor B. The shaded region represents (A U B) ′.

Ø  Fig.6 is (A ∩ B′) U (B ∩ A′) i.e. elements belonging to only A and only B. The shaded regions represent (A ∩ B′) U (B ∩ A′).

The mathematical expression connecting these regions is:

Where n (U) = n(A U B) + n(A U B)′.

Example 1

In a class of 36 students, 19 read Biology, 16 read Chemistry and 5 were not allowed to read both subjects. Illustrate the information on a Venn diagram. Find how many students read

1. Both Biology and Chemistry
2.  Only Chemistry
3. Only Biology
4. Only Biology and Only Chemistry

                                                       Solution

  Let U = {students in class}, this implies n (U) = 36.

Let B = {students who read Biology}, this implies n (B) = 19.

Let C = {students who read Chemistry}, this implies n(C) = 16.

Let X = {number of students who read both subjects}.

From the Venn diagram above:

1. 4 students read both subjects.
2. The number of students who  read only chemistry = 16-x

                                                                                     = 16-4

                                                                                     = 12.

3. The number of students who read only Biology = 19-x

                                                                                  = 19-4

                                                                                  = 15.

iv.   The number of students who read only Biology and only chemistry = (19-x) + (16-x)

                                                                                                                   =     15   +     12

                                                                                                                   = 27.

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