Venn Diagram in Set Theory

Venn diagrams usually represent a set. It is just the pictorial relationship among two or more sets where everything apart from the elements of the sets called universal set. Usually, universal set is represented by rectangular region, disjoint sets by disjoint circles and intersecting sets by intersecting circles. By universal sets, we mean the maximum number of possible points that can be included in any set.

Representation of Sets with Venn Diagrams

Case 1: When universal sets and a normal set have been given

Let U be the universal set representing the sets of all natural numbers and let X ⊆ U for X = 1,2,3,4,5

Then by Venn diagram we can show as:

Universal Set

Case 2: When two intersecting subsets of U are given

For representing two intersecting subsets A and B of U, we draw two circles and the intersecting regions will be shown as:

Sets Intersection

Case 3: When two disjoint sets are being given

To draw two disjoint sets, we normally draw two circles which will not intersect each other within a rectangle.

Disjoint Sets

Case 4: When B⊆A⊆U

In this case, we draw two concentric circles within a rectangular region

Concentric Sets

Case 5: Complement of a set

Let us set A and it is given as A = {1,2,3} i.e. A is a set of natural numbers.

Then the complement of A will be shown by:

Complementary Sets

We present the subset of one set by venn diagram as:

Let us take Y ⊆ X

If a ϵ Y and Y ⊆ X then it shows that a ϵ X also.

Solved Examples

Example 1: Consider the relations:

a) A – B = A – (A ∩ B)

b) A = (A ∩ B) ∪ (A – B)

c) A – (B ∩ C) = (A – B) ∪ (A – C)

Which of these options is/are correct?

  1. a and b
  2. b only
  3. b and c
  4. a and b

Answer:

A – B = A – (A ∩ B) is correct.

C is false.

So, a and b are true.

Option (d) is the correct answer.

Example 2: Which is the simplified representation of (A’∩ B’∩ C) ∪ (B ∩ C) ∪ (A ∩ C)

a) A

b) B

c) C

d) X ∩ (A ∪ B ∪ C)

Answer:

(A’∩ B’∩ C) = only C

(A’∩ B’ ∩ C) ∪ (B ∩ C) ∪ (A ∩ C) = C (by Venn Diagram)

venn diagram representation example

Example 3: In a class of 25 students, 12 students took singing,11 took dancing and 15 took writing; 4 took both singing and dancing; 9 took dancing and writing; 5 took writing and singing. 3 took all three. Find the number of students who took:

  1. Singing only;
  2. Dancing only;
  3. Writing only;
  4. Singing and dancing but not writing;
  5. writing and dancing but not singing;
  6. At least one of the three;
  7. None of the three

Answer:

Venn Diagram Representation Examples

Let the number of students in the respective regions be represented by a,b,c,d,e,f,g as shown above;

As per the question given the data are:

a + b + c + d = 12

b + c + e + f = 11

c + d + f + g = 15

b + c = 4

c + d = 9

c + f = 5

c = 3

By following these equations, we get:

c = 3; f = 2; d = 6; b = 1

Now,

c + d + f + g = 15 ⇒ 3 + 6 + 2 + g = 15 ⇒ g = 4

b + c + e + f = 11 ⇒ 1 + 3 + e + 2 = 11 ⇒ e = 5

a + b + c + d = 12 ⇒ a + 1 + 3 + 6 = 12 ⇒ a = 2

So, we have:

Number of students who took singing only = 2;

Number of students who took dancing only = 5;

Number of students who took writing only = 4;

Number of students who took singing and dancing but not writing = 1;

Number of students who took singing and writing but not dancing = 6;

Number Of students who took none = (25 – 23) = 2

Number of students who took only one = 2 + 5 + 4 = 11

Number of students who took at least one of the given subjects = 2 + 1 + 3 + 6 + 5 + 2 + 4 = 23.

 

 

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