Venn Diagram: The differences and similarities are visually represented when two concepts are compared and contrasted using a Venn diagram. A Venn diagram is also known as a logic diagram or a set diagram and is widely used in logic, mathematics, set theory, business, education, and computer science and engineering. Venn Diagrams area unit introduced by English logician John Venn (\left( {1834 – 1883} \right).) Venn’s diagram uses easy closed curves drawn on a plane to represent sets. It helps the child to visualize the problems of the set easily.

What is Venn Diagram?

Representation of relationships between sets by means of diagrams is known as Venn Diagrams.

The Venn Diagram consists of rectangles and closed curves, usually circles, sometimes ellipses. The universal set is represented normally by a rectangle and subsets of a universal set by circles or ellipses.

In Venn Diagrams, the elements of sets are commonly written in their respective circles.

Example: \(U = \left\{ {1,2,3,\,…10} \right\}\) is the universal set of which \(A = \left\{ {2,\,4,\,6,\,8,10} \right\}\) is a subset.

Union of Sets:

Operation on Sets Using Venn Diagrams

Several operations can be performed on sets using Venn Diagram, such as:

Let \(A\) and \(B\) be any two sets. The union of \(A\) and \(B\) is the set that consists of all the elements of \(A\) and all the elements of \(B\) the common elements being taken only once. The symbol \(U\) is used to denote the union.

Symbolically, we write \(A\, \cup \,B\) and usually read as \(A\) union \(B.\)

Example: Let \(A = \,\left\{ {4,\,6,\,8} \right\}\) and \(B = \,\left\{ {6,\,8,\,10,12,13,17} \right\}\) Find \(A\, \cup \,B\)
we have \(A \cup B = \left\{ {4,\,6,\,8,10,\,12,\,13,\,17} \right\}\)
The common elements \(6\) and \(8\) have been taken only once while writing \(A \cup B\)

Some Property of Operations of Union

1. \((A \cup B) = (B \cap A)\) (Commutative law)
2. \((A \cup B) \cup C = A \cup (B \cup C)\) (Associative law)
3. \((\phi \cup A) = A,(U \cup A) = U\) (Law of \(\phi \) and \(U\))
4. \((A \cup A) = A\) (Idempotent law)

Intersection of Sets

The intersection of sets \(A\) and \(B\) is the set of all elements which are common to both \(A\) and \(B\) The symbol \( \cap \) is used to denote the intersection. The intersection of two sets of two sets \(A\) and \(B\) is the set of all those elements which belong to both \(A\) and \(B\) Symbolically, we write \(A \cap B = \left\{ {x:\,\,x \in A\,and\,x\, \in \,B} \right\}\)

Example: Let \(A = \,\left\{ {2,\,\,4,\,\,6,\,8} \right\}\) and \(B = \,\left\{ {6,\,\,8,\,\,10,\,12} \right\}\) Find \(A \cap B\) We see that \(6,8\) are the only elements that are common to both \(A\) and \(B\). Hence, \(A\, \cap B = \left\{ {6,\,8} \right\}\)

Some property of operations of intersection:

1. \(A\, \cap B = (B \cap A)\) (Commutative law)
2. \((A \cap B) \cap C = A \cap (B \cap C)\) (Associative law)
3. \((\phi \cap A) = \phi ,(U \cap A) = A\) (Law of \(\phi \) and U)
4. \((A \cap A) = A\) (Idempotent law)
5. \(A \cap (B \cup C) = (A \cap B) \cup (B \cap C)\) (Distributive law)

Complement of a Set:

Let \( \cup \) be the universal set, and \(A\) be a subset \( \cup \) Then, the complement of \(A\) is the set of all elements of \( \cup \) which are not the elements of \(A\).
Symbolically, we use \(A’\) to represent the complement of \(A\) with respect to U.
Thus,
\(A’\, = \,\left\{ {x:\,x\, \in U\,and\,x \notin A} \right\}.\) Obviously, \(A’\, = \,U – A\)
We know that the complement of a set \(A\) can be looked upon, alternatively, as the difference between a universal set \(U\) and set \(A\)

Example: Let \(U = \left\{ {1,2.3,4,5,6,7,8,9,10} \right\}\) and \(A = \left\{ {1,\,3,\,5,\,7,\,9} \right\}\) Find \(A’\).
From the given data, we observed that \(2,\,4,\,6,\,8,\,10\) are the elements of \(U\) which do not belong to set \(A\).
So, \(A’ = \left\{ {2,4,\,6,\,8,\,10} \right\}\)

Disjoint Sets

If \(A\) and \(B\) are two sets such that \(A \cap B\, = \,\phi \) then \(A\) and \(B\) are called disjoint sets.

Example:  let \(A = \,\left\{ {2,4,\,6,\,8} \right\}\,\,and\,B = \left\{ {1,3,\,5,\,7} \right\}\) Then \(A\) and \(B\)
are disjoint sets because there are no common elements in set \(A\) and set \(A\)

Difference of Sets

The difference of set \(A\) and \(B\) in this order is the set of elements which belongs to \(A\) but not to \(B\).
Symbolically, we write as \(A\,\, – \,\,B\)
Examples: Let \(A = \left\{ {2,\,3,\,4,\,6,\,8} \right\}\) and \(B = \left\{ {6,\,4,\,5,\,11,\,14} \right\}\) Find\(A\,\, – \,\,B\)
The difference of set \(A\) and \(B\) in this order is the set of elements that belongs to set \(A\) but not to set \(B\).
Therefore, \(A\, – \,B = \left\{ {2,\,3,\,8} \right\}\)

What Is the Formula for Venn Diagram?

1. \(n(A \cup B) = n(A) + n(B) – n(A \cap B)\)
2. \(n(A \cup B \cup C) = n(A) + n(C) – n(A \cap B) – n(B \cap C) – n(C \cap A) + n(A \cap B \cap C)\)

Where \(n(A) \to \) the number of elements in set \(A\).
\(n(B) \to \) the number of elements in set \(B\).
\(n(C) \to \) the number of elements in set \(C\).

Venn Diagram Symbols

The symbols used to represent the operations of sets are:
Union of sets symbol: \( \cup \)
The intersection of sets symbol: \( \cap \)
Complement of set: \(A’\)
Any set can be represented by a circle, and a universal set is represented by a rectangle.

Types of Venn Diagrams

Different types of Venn Diagrams are,

1. Two-set Venn Diagrams: These Venn Diagrams consists of two circles or ovals to show overlapping properties.

2. Three-set Venn Diagrams: In a three-set Venn Diagram, all sets have some overlap with each other.

3. Four-set Venn Diagrams: These Venn Diagrams consists of four ovals to show overlapping properties.

4. Five-set Venn Diagrams: These Venn Diagrams consists of five ovals to show overlapping properties.

Uses of Venn Diagrams

1. Venn Diagram is an example that uses circles to show the relationships among finite groups of elements.
2. Venn Diagrams are used both for comparison and classification.
3. Venn Diagrams help to group the information into different chunks.
4. Venn Diagrams categorize and classify information.
5. Venn Diagrams highlight similarities and differences.
6. Venn Diagrams are used in different fields such as linguistics, business, statistics, logic, mathematics, teaching, computer science etc.
7. Venn Diagram can be used for analyzing the effectiveness of websites.
8. Venn Diagrams are used in wellbeing and Psychology.
9. Venn Diagrams are used in mathematics to divide all possible number types into groups.
10. Venn Diagrams help visually represent the similarities and difference between two concepts.

Solved Examples – Venn Diagram

Q.1. Let \(A = \left\{ {2,\,4,\,6,8} \right\}\) and \(B = \left\{ {6,\,\,8,\,\,10,\,12} \right\}\). Find \(A\; \cup B.\)
Ans: From the given \(A = \left\{ {2,\,4,\,6,8} \right\}\) and \(B = \left\{ {6,\,\,8,\,\,10,\,12} \right\}\)
Therefore, \(A \cup B = \left\{ {2,\,4,\,6,\,8,\,10,\,12} \right\}.\)

Q.2.  Let \(A = \left\{ {a,\,e,\,i,\,o,\,u} \right\}\) and \(B = \left\{ {a,\,r,\,i,\,g} \right\}\) Find \(A \cap B\)
Ans: From the given \(A = \left\{ {a,\,e,\,i,\,o,\,u} \right\}\) and \(B = \left\{ {a,\,r,\,i,\,g} \right\}\).
Therefore, \(A \cap B = \left\{ {a,\,i} \right\}.\)

Q.3.  Let \( \cup \, = \,\left\{ {1,\,2,\,3,\,4,\,5,\;6,\,7,\,8,\,9,\,10} \right\}\,\,and\,A = \,\left\{ {1,\,2,\,6,\,7,\,9} \right\}\) Find \(A’\).
Ans: From the given \(U = \left\{ {1,2,\,3,\,4,\,5,\,6,\,7,\,8,\,9,\,10} \right\}\,and\,A\, = \,\left\{ {1,\,2,\,6,\,7,\,9} \right\}\)
We note that \(3,\,4,\,5,\,8,\,10\) are the only elements of \(U\) which do not belong to \(A\).
Therefore, \(A’ = \left\{ {3,\,4,\,5,\,8,\,10} \right\}.\)

Q.4. Let \(A = \left\{ {2,\,\,5,\,6,\,8} \right\}\,and\,B = \left\{ {6,\,8,\,11,\,14} \right\}\) Find \(A \cap B\)
Ans: We see that \(6,\,\,8\) are the only elements that are common to both \(A\) and \(B\).
Hence, \(A \cap B\, = \,\left\{ {6,\,\,8} \right\}.\)

Q.5. Let \(A = \left\{ {2,\,\,5,\,6,\,8} \right\}\,and\,B = \left\{ {6,\,8,\,11,\,14} \right\}\) Find \(A\,\, – \,\,B\).
Ans: From the given, \(A = \left\{ {2,\,\,5,\,6,\,8} \right\}\,and\,B = \left\{ {6,\,8,\,11,\,14} \right\}\)
The difference of set \(A\) and \(B\) in this order is the set of elements which belongs to \(A\) but not to \(B\)
Therefore, \(A – B = \left\{ {2,\,5} \right\}.\)

Q.6. If \(X\, and \,Y\) are two sets such that \(X\, \cup \,Y\) has \(50\) elements, \(X\) has \(28\) elements, and \(Y\) has \(32\) elements, how many elements does \(X\, \cap \,Y\)?
Ans: From the given \(n(X) = 28,\,n(Y) = 32,\,n(X \cup Y) = 50,n(X \cap Y) = ?\)
By using the formula \(n(X \cup Y) = n(X) + n(Y) – n(X \cap Y)\)
\( \Rightarrow 50 = 28 + 32 – n(X \cap Y)\)
\( \Rightarrow n(X \cap Y)\, = \,60\, – \,50\, = 10\)
Therefore, \((X \cap Y)\) have \(10\) elements.
Alternative method:
Let \(n(X \cap Y)\, = \,k,\) then \(n(X – Y)\, = \,28 – k,\,n(Y – X) = 32 – k,\,n(X \cup Y) = 50\)
\( \Rightarrow \left( {X \cup Y} \right) = n(X – Y) + n(Y – X) + n(X \cap Y)\)
\( \Rightarrow 50 = (28 – k) + (32 – k) + k\)
\( \Rightarrow 50 = 28 – k + 32 – k + k\)
\( \Rightarrow k = 60 – 50\)
\( \Rightarrow k = 10\)
Therefore, \((X \cap Y)\) have \(10\) elements.

Summary

In this article, we have discussed that Venn’s diagram is a pictorial exhibition of all attainable real relations between a group of varied sets of things. It is created from many overlapping circles or oval shapes, with every circle representing one set of items. This article helps students to learn in detail about Venn Diagram, symbols, operations of sets on Venn Diagram. At the end, we have discussed some solved examples which will help the child to understand this topic in a better way.

Frequently Asked Questions (FAQs) About Venn Diagram

Let’s look at some of the frequently asked questions about Venn Diagrams:

Q.1. What are the different types of Venn Diagrams?
Ans: Different types of Venn Diagram are,
a. Two-set Venn Diagrams.
b. Three-set Venn Diagrams.
c. Four-set Venn Diagrams.
d. Five-set Venn Diagrams.

Q.2. What are \(3\) intersecting circles called?
Ans: \(A3\)- circle Venn Diagram, named after the English logician Robert Venn, is a diagram that shows how the elements of three sets are related using three overlapping circles.

Q.3. What is the complement of a Set?
Ans: The complement of a set is represented by \(A’\) is the set of all elements in the given universal set that are not in \(A\).

Q.4. What does \(A \cap B\) represent?
Ans: \(A \cap B\) represents the intersection of two sets \(A\) and \(B\). \(A \cap B\) is the set of all the common elements which belong to both \(A\) and \(B\).

Q.5. What is a subset in a Venn Diagram?
Ans: A subset is a set that is entirely contained within another set. Each set in a Venn Diagram is a subset of that diagram.

Q.6. Write Venn Diagram formulas?
Ans: The formulas of the Venn Diagram are,
1. \(n(A \cup B) = n(A) + n(B) – n(A \cap B)\)
2. \(n(A \cup B) = n(A) + n(B) + n(C) – n(A \cap B) – n(B \cap C) – n(C \cap A) + n(A \cap B \cap C)\)
Where \(n(A) \to \) represents the number of elements in set \(A\).
\(n(B) \to \) the number of elements in set \(B\).
\(n(C) \to \) the number of elements in set \(C\).

Q.7. What is Venn Diagram?
Ans: Most of the relationships between sets can be represented by means of diagrams which are known as Venn Diagrams. The Venn Diagram consists of rectangles and closed curves, commonly circles. The universal set is represented commonly by a rectangle and its subsets by circles.

Q.8. What are the uses of Venn Diagrams?
Ans: Following are the uses of Venn Diagram:

1. Venn Diagram is an example that uses circles to show the relationships among finite groups of elements.
2. Venn Diagrams are used both for comparison and classification.
3. Venn Diagrams help to group the information into different chunks.
4. Venn Diagrams categorize and classify information.
5. Venn Diagrams highlight similarities and differences.
6. Venn Diagrams are used in different fields such as linguistics, business, statistics, logic, mathematics, teaching, computer science etc.

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