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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 and Operations on Sets | Algebra - MathematicsVenn Diagrams and Operations on Sets | Algebra - Mathematics

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 and Operations on Sets | Algebra - Mathematics

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 U – 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:

Venn Diagrams and Operations on Sets | Algebra - Mathematics

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 and Operations on Sets | Algebra - Mathematics

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 and Operations on Sets | Algebra - Mathematics

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 and Operations on Sets | Algebra - Mathematics

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

Solved Examples For You

Ques 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.

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

Venn Diagrams and Operations on Sets | Algebra - Mathematics

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:

a. A-B

Venn Diagrams and Operations on Sets | Algebra - Mathematics

b. B-A 

Venn Diagrams and Operations on Sets | Algebra - Mathematics

Operations on sets :
When two or more sets are combined together to form another set under some given conditions, then operations on sets are carried out.
The important operations on sets are. 
Union
Let X and Y be two sets.
Now, we can define the following new set.
X u Y  =  {z | z ∈ X or z ∈ Y}
(That is, z may be in X or in Y or in both X and Y) 
X u Y is read as "X union Y"
Now that X u Y contains all the elements of X and all the elements of Y and the figure given below illustrates this. 

Venn Diagrams and Operations on Sets | Algebra - Mathematics

It is clear that X ⊆ X u Y and also Y ⊆ X u Y
Intersection
Let X and Y be two sets.
Now, we can define the following new set.
X n Y  =  {z | z ∈ X and z ∈ Y}
(That is z must be in  both X and Y)
X n Y is read as "X intersection Y"
Now that X n Y contains only those elements which belong to both X and Y and the figure given below illustrates this. 

Venn Diagrams and Operations on Sets | Algebra - Mathematics

It is trivial that that X n Y ⊆ X and also X n Y ⊆ Y
Set difference
Let X and Y be two sets.
Now, we can define the following new set.
X \ Y  =  {z | z ∈ X but z ∉  Y}
(That is z must be in  X and must not be in Y)
X \ Y is read as "X difference Y"
Now that X \ Y contains only elements of X which are not in Y and the figure given below illustrates this. 

Venn Diagrams and Operations on Sets | Algebra - Mathematics

Some authors use A - B for A \ B. We shall use the notation A \ B which is widely used in mathematics for set difference. 
Symmetric difference
Let X and Y be two sets.
Now, we can define the following new set.
X Δ  Y  =  (X \ Y) u (Y \ X)
Δ Y is read as "X symmetric difference Y"
Now that X Δ Y contains all elements in X u Y which are not in X n Y and the figure given below illustrates this. . 

Venn Diagrams and Operations on Sets | Algebra - Mathematics

Complement
If X ⊆ U, where U is a universal set, then U \ X is called the compliment of X with respect to U.
If underlying universal set is fixed, then we denote U \ X by X' and it is called compliment of X.
X'  =  U \ X The difference set set A \ B can also be viewed as the compliment of B with respect to A.  

Venn Diagrams and Operations on Sets | Algebra - Mathematics

Disjoint sets
Two sets X and Y are said to be disjoint if they do not have any common element. That is,
X and Y are disjoint if X n Y = ᵩ
It is clear that n(A u B) = n(A) + n(B), if A and B are disjoint finite set.

Venn Diagrams and Operations on Sets | Algebra - Mathematics

The document Venn Diagrams and Operations on Sets | Algebra - Mathematics is a part of the Mathematics Course Algebra.
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FAQs on Venn Diagrams and Operations on Sets - Algebra - Mathematics

1. What is a Venn diagram and how is it used in mathematics?
A Venn diagram is a visual representation of sets using overlapping circles or other shapes. It is commonly used in mathematics to illustrate relationships between different sets or groups of objects. The overlapping portions of the circles represent elements that belong to multiple sets, while the non-overlapping portions represent elements that belong to only one set.
2. What are the main operations performed on sets using Venn diagrams?
The main operations performed on sets using Venn diagrams include union, intersection, and complement. - Union: The union of two sets is represented by the combination of all elements from both sets. In a Venn diagram, it is shown by the overlapping region of the circles representing the sets. - Intersection: The intersection of two sets is the set of elements that are common to both sets. In a Venn diagram, it is represented by the overlapping portion of the circles. - Complement: The complement of a set is the set of elements that do not belong to that particular set. In a Venn diagram, it is shown by the region outside the circle representing the set.
3. How can Venn diagrams be used to solve problems involving set operations?
Venn diagrams provide a visual representation of sets and their relationships, making it easier to understand and solve problems involving set operations. By using the diagram, one can identify the common elements between sets (intersection), the combined elements of sets (union), and the elements that do not belong to a set (complement). Venn diagrams can help in organizing information and visualizing complex relationships, leading to a better understanding of set operations.
4. Can Venn diagrams be used for more than two sets?
Yes, Venn diagrams can be used for more than two sets. In addition to the two-circle Venn diagram commonly used for two sets, there are also three-circle and even more complex Venn diagrams that can represent relationships between multiple sets. These diagrams have overlapping regions to show elements belonging to multiple sets and can be useful in solving problems involving multiple set operations.
5. Are there any limitations to using Venn diagrams?
While Venn diagrams are a helpful tool for visualizing sets and their operations, they do have some limitations. One limitation is that they can become cluttered and difficult to interpret when dealing with a large number of sets or complex relationships. Another limitation is that Venn diagrams may not accurately represent the cardinality (number of elements) of each set. In such cases, other methods like set notation or mathematical equations may be more precise. Nonetheless, Venn diagrams remain a valuable tool for introducing and understanding set operations.
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