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Sets

Venn Diagrams

Thus far we have had a lot of fun with the concept of sets and their various types and applications. But it has just started. These diagrams were first introduced by John Venn as a tool to study logic. It turns out that these Venn Diagrams can also be used to study and represent sets. The fun is that these diagrams make working with sets easier and fun too! Let us see more.

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Venn Diagrams

A Venn diagram is a diagrammatic representation of ALL the possible relationships between different sets of a finite number of elements. Venn diagrams were conceived around 1880 by John Venn, an English logician, and philosopher. They are extensively used to teach Set Theory. A Venn diagram is also known as a Primary diagram, Set diagram or Logic diagram.

Venn Diagrams

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Representation of Sets in a Venn Diagram

It is done as per the following:

  • Each individual set is represented mostly by a circle and enclosed within a quadrilateral (the quadrilateral represents the finiteness of the Venn diagram as well as the Universal set.)
  • Labelling is done for each set with the set’s name to indicate difference and the respective constituting elements of each set are written within the circles.
  • Sets having no element in common are represented separately while those having some of the elements common within them are shown with overlapping.
  • The elements are written within the circle representing the set containing them and the common elements are written in the parts of circles that are overlapped.

Operations on Venn Diagrams

Just like the mathematical operations on sets like Union, Difference, Intersection, Complement, etc. we have operations on Venn diagrams that are given as follows:

Union of Sets

Let A = {2, 4, 6, 8} and B = {6, 8, 10, 12}. Represent A U B through a well-labeled Venn diagram.

Venn Diagrams
The orange colored patch represents the common elements {6, 8} and the quadrilateral represents A U B.

Properties of A U B

  • The commutative law holds true as A U B = B U A
  • The associative law also holds true as (A U B) U C = A U (B U C)
  • A U φ = A (Law of identity element)
  • Idempotent Law – A U A = A
  • Law of the Universal Set – A U U = U

The Intersection of Sets

An intersection is nothing but the collection of all the elements that are common to all the sets under consideration. Let A = {2, 4, 6, 8} and B = {6, 8, 10, 12} then A ∩ B is represented through a Venn diagram as per following:

The orange colored patch represents the common elements {6, 8} as well as the A ∩ B. The intersection of 2 or more sets is the overlapped part(s) of the individual circles with the elements written in the overlapped parts. Example:

Venn Diagrams

Properties of A ∩ B

  • Commutative law – A ∩ B = B∩ A
  • Associative law – (A ∩ B)∩ C = A ∩ (B∩ C)
  • φ ∩ A = φ
  • U ∩ A = A
  • A∩ A = A; Idempotent law.
  • Distributive law – A ∩ (B∩ C) = (A ∩ B) U(A ∩ C)

Difference of Sets

The difference of set A and B is represented as: A – B = {x: x ϵ A and x ϵ B} {converse holds true for B – A}. Let, A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8} then A – B = {1, 3, 5} and B – A = {8}. The sets (A – B), (B – A) and (A ∩ B) are mutually disjoint sets.

It means that there is NO element common to any of the three sets and the intersection of any of the two or all the three sets will result in a null or void or empty set. A – B and B – A are represented through Venn diagrams as follows:

Venn Diagrams

Complement of Sets

If U represents the Universal set and any set A is the subset of A then the complement of set A (represented as A’) will contain ALL the elements which belong to the Universal set U but NOT to set A.

Mathematically – A’ = U – A

Alternatively, the complement of a set A, A’ is the difference between the universal set U and the set A. Example: Let universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and set A = {1, 3, 5, 7, 9}, then complement of A is given as: A’ = U – A = {2, 4, 6, 8, 10}

Venn Diagrams

Properties Of Complement Sets

  • A U A’ = U
  • A ∩ A’ = φ
  • De Morgan’s Law – (A U B)’ = A’ ∩ B’ OR (A ∩ B)’ = A’ U B’
  • Law of double complementation : (A’)’ = A
  • φ’ = U
  • U’ = φ

Word Problems Using Venn Diagram

Solved Examples For You

Question 1: Represent the Universal Set (U) = {x : x is an outcome of a dice’s roll} and set A = {s : s ϵ Even numbers} through a Venn diagram.

Answer : Since, U = {1, 2, 3, 4, 5, 6} and A = {2, 4, 6}. Representing this with a Venn diagram we have:

Venn Diagrams

Here, A is a subset of U, represented as – A ⊂ U or
U is the superset of A, represented as – U⊃ A
If A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8},
then represent A – B and B – A through Venn diagrams.
A – B = {1, 2, 3}
B – A = {6, 7, 8}
Representing them in Venn diagrams:

Venn Diagrams

Question 2: Define the working of the Venn diagrams.

Answer: Venn diagrams permit the students to arrange the information visually so that they are able to see the relations between 2 or 3 sets of the items. They can then recognize the similarities and differences between them. A Venn diagram figure contains overlapping circles. Each circle comprises all the elements of one set.

Question 3: How can we describe a Venn diagram?

Answer: A Venn diagram is made with the overlapping circles. The inner of each circle shows a set of objects, or objects having something in common. The outside of the circle symbolizes all that a single set excludes.

Question 4: What is the medium of a Venn diagram known as?

Answer: The area of the joining of the 3 circles is the medium or middle point of a Venn diagram.

Question 5: What does the ‘U’ mean in a Venn diagram?

Answer: The ‘∪’ in a Venn diagram represents the ‘union of 2 sets’. Each sphere or ellipse in a Venn diagram represents a group.

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sidharth

sir can we prove all the set theorems using venn diagram ?

Ateek kumar
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Ateek kumar

No only few sets can be define

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