Venn diagrams

To represent sets, we use the roster form or the set-builder form. To represent the relations and operations among the sets, we use a Venn diagram. The name is given after John Venn (1834-1883). The salient features of Venn diagrams are:

• The universal set (U) is usually represented by a closed rectangle. However, it can be represented by any closed curve.
• Any subset of the universal set is represented by closed curves, generally represented by circles.

Note: It is to be noted that Venn diagram can be represented by any closed figure, whether it be a Circle or a Polygon (square, hexagon, etc.).

How to draw a Venn diagram?

To draw a Venn diagram, first, the universal set should be decided. Now, every set is the subset of the universal set (U). This means that every other set will be inside the rectangle which represents the universal set.

So, any set A (shaded region) will be represented as follows:

Figure 1: Set A

We can say from the fig. 1 that

$A \cup U = U$<

All the elements of set A, are inside the circle. Also, they are part of the big rectangle which makes them the elements of set U. The representations of different operations on a set are as follows:

• A’: This is the complement of set A (represented by the shaded region in fig. 2). This set contains all the elements which are not there in set A.

Figure 2: A’

It is visually clear that

A+A’=U

(A’ )’= A

It means that the set formed with elements of set A and set A’ combined is equal to U.

• A∩B: This is read as A intersection B. This represents the common elements between set A and B (represented by the shaded region in fig. 3).

Figure 3: A∩B

• A∪B: This is read as A union B. This represents the combined elements of set A and B (represented by the shaded region in fig. 4).

Figure 4: A∪B

• (A∪B)’: This is read as complement of A union B. This represents elements which are neither in setA nor in setB (represented by the shaded region in fig. 5).

Figure 5: (A∪B)’

• (A∩B)’: This is read as complement of A intersection B. This represents elements of the universal set which are not common between set A and B (represented by the shaded region in fig. 6).

Figure 6: (A∩B)’

• A-B: This is read as A difference B. Sometimes, it also referred as ‘relative complement’. This represents elements of set A which are not there in set B(represented by the shaded region in fig. 7).

Figure 7: A-B

• A⊝B: This is read as symmetric difference of set A and B. This is a set which contains the elements which are either in set A or in set B but not in both (represented by the shaded region in fig. 8).

Figure 8: A ⊝ B

Venn diagrams: Sample Questions

Example: In a class of 50 students, 10 take Guitar lessons and 20 take singing classes, and 4 take both. Find the number of students who don’t take either Guitar or singing lessons.

Solution:

Let A = no. of students who take guitar lessons = 10.

Let B = no. of students who take singing lessons = 20.

Let C = no. of students who take both = 4.

Now we subtract the value of C from both A and B. Let the new values be stored in D and E.

Therefore,

D = 10 – 4 = 6

E = 20 – 4 = 16

Now logic dictates that if we add the values of C, D, E and the unknown quantity “X”, we should get a total of 50 right? That’s correct.

So the final answer is X = 50 – C – D – E

X = 50 – 4 – 6 – 16