Venn Diagram

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What is a Venn Diagram?

Venn diagram is a diagram that represents the relationship between and among a finite group of sets. Venn diagram was introduced by John Venn around 1880. These diagrams are also known as set diagrams or logic diagrams showing different sets of operations such as the intersection of the set, union of the set, and difference of sets. It is also used to represent subsets of a set.

For example, a set of natural numbers is a subset of whole numbers, which is a subset of integers. The relationship between a set of natural numbers, whole numbers, and integers can be represented by the Venn diagram as shown below. 

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In the above Venn diagram, the set of integers represents the universal set, and the set of natural numbers (N) is a subset of the whole number (W). The universal set is usually represented by a closed rectangle, consisting of all the sets. The sets and subsets are represented by using circles or oval.


Venn Diagram Definition

A  Venn diagram is a diagram that is used to represent all the possible relations of different sets. A Venn diagram can be represented by any closed figure whether it be a circle or polygon. Generally, circles are used to denote each set.

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In the above Venn diagram, we can see that the rectangular universal set includes two independent sets X and Y. Therefore, X and Y are considered as disjoint sets. The two disjoint sets X and Y are represented in a  circular shape. The above Venn diagram states that X and Y have no relation with each other, but they are part of a universal set. 

For example, set X = { set of multiple of 5}, and set Y = { set of multiple of 7} and Universal set  U = { set of natural numbers}


Drawing Venn Diagram

To draw a Venn diagram, we first draw a rectangle that will include every item that we want to consider. As it includes every item, we can refer to it as ‘ the universal set’. As we know, each set is the subset of the universal set (U). It means that every other set that is drawn inside the rectangle represents the universal set.

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For example, the Venn diagram given above represents any set A in a shaded region.

Where U is the universal set 

The above Venn diagram states that A ∪ U = U. It means all the elements of set A are inside the circles. Also, they are part of the big rectangle which makes them the elements of the Universal set (U).


Venn Diagram Formulas

Some basic venn diagram formula of 2 or 3 elements are discussed below:

  • n ( X ∪Y) = n (X) + n(Y) - n( X ∩ Y)

  • n ( X ∪ Y ∪ Z) = n(X) + n(Y) + n(Z) - n( X ∩ Y) - n( Y ∩ Z) - n ( Z ∩ X ) + n( X  ∩ Y ∩ Z) 

n(X) in the above Venn diagram formula represents the number of elements in set X.


Venn Diagram For 2 Sets

Venn diagram for 2 sets i.e. n ( A ∪B) = n (A) + n(B) - n( A ∩ B) is shown below:

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In the above Venn diagram, 

  • A represents the number of elements belonging to set A only.

  • B represents the number of elements that belong to set B only

  • A & B represent the number of elements that belong to both set A and B

  • A or B represents the set of all the elements belonging to A or to B.

  • U represents the universal set that includes all the elements or objects of other sets including its own elements.


Venn Diagram For 3 Set

The intersection of three sets X, Y, and Z is the group of elements that are included in all the three sets X, Y, and Z. It is represented by  X ∩ Y ∩ Z. Let us understand the Venn diagram for 3 sets with an example below.

Example:

Construct a Venn diagram to denote the relationships between the following sets i.e.

X = { 1,2,5,6,7,9} Y = ( 1,3,4,5,6,8} and Z = { 3,5,6,7,8,10}

Solution :

We find that X ∩ Y ∩ Z = { 5,6} X ∩ Y = { 1,5,6} ,Y ∩ Z = { 3,5,6,8}, and X ∩ Z= { 5,6,7}


Steps to Represent the Above Relations by Drawing a Venn Diagram for 3 Sets X, Y, and  Z

  1. Draw three intersecting circles to represent the given three sets.

  2. First, fill all elements that should be included in the intersection X ∩ Y ∩ Z

  3. Write down the remaining elements in the intersection X ∩ Y,Y ∩ Z, and X  ∩ Z.

  4. The elements that are left at last should be included in the respective sets.

Note: It is suggested to fill the Venn diagram with all the possible elements that are intersecting in 2 or more than 2 sets as shown below.

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Universal Set Venn Diagram

A universal set is defined as a set that incorporates all the elements or objects of other sets including its own elements. It is usually represented by the symbol ‘U’.

For the representation of the Venn diagram universal set, we can consider an example as Set U={heptagon},  Set A={pentagon, hexagon, octagon} , and Set C={nonagon}.

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Note: There is no formula to determine the universal set, we simply have to represent all the elements in a single set which is collectively known as the universal set. Even, there is no standard symbol used to represent a universal set. The common universal set symbol includes V, U, and ६.


Real Number Venn Diagram

In Mathematics, set is an ordered group of objects and can be denoted in a set builder form or roster form. Generally, sets are denoted in curly braces {}. For example, A = { 1,2,3,4} is a set.

The set of real numbers includes the set of rational numbers and the set of irrational numbers. We can represent the set of real numbers using a Venn diagram as shown below. The real number Venn diagram shown below clearly illustrates the set of real numbers. 

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Venn Diagram Subsets

Although Venn diagrams are commonly used to represent intersection, union, and complements of sets, they can also be used to represent subsets.

A subset is a set that is completely included in another set. In other words, If every element in one set is included in another set, they are called a subset. For example, the Venn diagram subset shown below states that A is a subset of B.

A is the subset of B, therefore, we will draw a small circle A inside the big circle B. A ∩ B implies that we have to shade the common portion of A and B.

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Hence, the above Venn diagram clearly states that A ∩ B = A


Venn Diagram Examples

  1. Answer the Following Based on the Venn Diagram Given Below

List the elements of

U

Aˡ  ∪ Bˡ

Aˡ  ∪  Bˡ

Solutions:

  1. U = { 3,4,5,7,8,11,12,13}

  2. Aˡ = { 4,7,8,9,11,13}

  3. Bˡ = { 5,7,11,12,13}

  4. Aˡ ∪Bˡ = {4,5,7,8,9,11,12,13}

  5. Aˡ ∪ Bˡ = {7,11,13}


  1. What Does the Shaded Portion in the Venn Diagram Given Below Denote?

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Solution: The 3 circle Venn diagram represents the common elements between A, B, and C. Hence, the above 3 circles venn diagram shows A ∩ B ∩ C.

FAQ (Frequently Asked Questions)

1. What is set in Mathematics?

Ans: In Mathematics, a set is a group of well defined distinct objects represented in the set-builder form or roster form. The well defined refers to the specific features that make it easy to identify whether the given object belongs to a set or not. The word distinct implies that the objects included in the set must all be different.

2. What are the different uses of Venn diagrams?

Ans:
  • Venn diagrams are commonly used in schools to make the students understand the Mathematics concepts like sets, the intersection of sets, and the union of sets. Venn diagrams are also used to study the similarities and differences among different languages.

  • Venn diagrams can be used by programmers to envisage the computer languages and hierarchies.

  • Venn diagrams are used by the statistician to estimate the possibilities of certain occurrences.

  • Teachers can make use of Venn diagrams to enhance the student’s reading comprehension. Students can construct Venn diagrams to compare the similarities and differences between the ideas they are reading.

3. Who introduced Venn diagrams?

Ans: Venn diagram was introduced by the English Logician John Venn in the 1880s. He gave the name Eulerian Circles after the Swiss Mathematician Leonard Euler, who introduced similar diagrams in the 1770s.

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