Overview Venn Diagram - General Intelligence and Reasoning for SSC CGL

Venn Diagram

Mathematics often requires clear representation of relationships between groups of items. A Venn diagram is a pictorial method that organises and displays such relationships using simple closed curves - usually circles - drawn inside a rectangle that represents the Universal set. Overlapping regions of the circles show common elements, while non-overlapping regions show elements unique to a particular set. Venn diagrams make classification, comparison and reasoning about sets straightforward and are especially useful in logical reasoning and data interpretation.

What is a Venn Diagram?

A Venn diagram represents sets as circles within a rectangular frame that denotes the Universal set. When two or more circles overlap, the overlapping area denotes elements common to the corresponding sets; where they do not overlap, the elements are distinct to that set. The diagrams are named after the logician John Venn, who introduced this form of diagrammatic representation in 1918.

Key advantages of Venn diagrams:

• They classify data that belong to the same broad category but different sub-categories.
• They make comparison and visualisation of relationships between groups easier.
• They help in grouping information and identifying similarities and differences quickly.
• They assist in finding unknown parameters by visual inspection and by translating the diagram into set formulae.

Example of a Venn Diagram

Example: Take a set A representing even numbers up to 10 and another set B representing natural numbers less than 5 then their interaction is represented using the Venn diagram.

Solution:

Terms Related to Venn Diagram

Understanding a few standard terms simplifies working with Venn diagrams and set operations.

Universal Set

The Universal set (usually denoted by U) is the set that contains all elements under consideration for a particular problem. Every set discussed in that context is a subset of U. For example, if set A is the set of Honda cars in a society and set B is the set of red cars in the same society, then the set of all cars in that society is the Universal set because it contains all elements of A and B.

Subset

A set B is a subset of set A (written B ⊆ A) when every element of B is also an element of A. Example: Let N be the set of natural numbers and W the set of whole numbers. Then N ⊆ W because every natural number is also a whole number.

Venn Diagram Symbols

Before drawing Venn diagrams it is helpful to be familiar with the standard set symbols and their meanings. Essential symbols used with Venn diagrams include ∪ (union), ∩ (intersection), A' or Aᶜ (complement), ∅ (empty set) and ⊆ (subset). These symbols help convert a diagram into algebraic set expressions and vice versa.

How to Draw a Venn Diagram

Follow these simple steps to draw a Venn diagram appropriate to the number of sets and the relationships given in a problem.

1. Draw a rectangle to represent the Universal set U.
2. Draw one circle for each set. Position the circles so that they overlap where sets have common elements and remain separate where they have no common elements.
3. Mark or compute the elements in each distinct and overlapping region according to the given information; use set operations to find unknown counts.

Venn Diagram for Set Operations

The main set operations represented using Venn diagrams are:

• Union of sets
• Intersection of sets
• Complement of a set
• Difference of sets

Each operation corresponds to a particular region or combination of regions in the diagram; visual inspection often gives the answer directly and supports formulation of algebraic expressions.

Union of Sets

The union of sets A and B, written A ∪ B, is the set of elements that belong to A or to B or to both. Formally:

n(A ∪ B) = { a : a ∈ A or a ∈ B }

Important properties of union:

• A ∪ B = B ∪ A (commutative)
• A ∪ (B ∪ C) = (A ∪ B) ∪ C (associative)
• A ∪ U = U
• A ∪ Aᶜ = U
• A ∪ A = A

Intersection of Sets

The intersection of sets A and B, written A ∩ B, is the set of elements common to both A and B. Formally:

n(A ∩ B) = { a : a ∈ A and a ∈ B }

Important properties of intersection:

• A ∩ B = B ∩ A (commutative)
• (A ∩ B) ∩ C = A ∩ (B ∩ C) (associative)
• A ∩ A = A
• A ∩ U = A
• A ∩ Aᶜ = ∅

Complement of a Set

The complement of set A (denoted Aᶜ or A') consists of all elements in the Universal set U that are not in A.

n(Aᶜ) = n(U) - n(A)

Difference of Sets

The difference A - B (also written A \ B) is the set of elements that belong to A but not to B; on a Venn diagram it is the portion of A outside the overlap with B.

Example: If A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8} then A - B = {1, 3, 5}.

Types of Venn Diagrams

Venn diagrams are classified by the number of sets (circles) shown:

• Two-set Venn diagram
• Three-set Venn diagram
• Four-set Venn diagram
• Five-set Venn diagram

Venn Diagram for Three Sets

Three sets are shown by three overlapping circles, producing seven distinct non-empty regions plus possibly an region outside all circles (if elements exist in U that belong to none of the three sets). This lets us read off counts such as elements common to exactly two sets, exactly three sets, or to only one set.

Consider sets A = people who play cricket, B = people who are graduates, and C = people aged 18 or above. The three-set Venn diagram displays all possible overlaps among A, B and C and allows direct identification of regions such as A ∩ B, A ∩ B ∩ C, and so on.

Examples of quantities that can be identified from a three-set diagram:

• Number of graduates who play cricket: B ∩ C
• Number of graduates who play cricket and are at least 18 years old: A ∩ B ∩ C

Venn Diagram Formula

Set formulas translate diagram regions into algebraic expressions used for counting.

For two sets A and B:

\( n(A \cup B) = n(A) + n(B) - n(A \cap B) \)

where n(X) denotes the number of elements in set X.

For three sets A, B and C:

\( n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(C \cap A) + n(A \cap B \cap C) \)

Example: In a class of 40 students, 18 like Mathematics, 16 like Science, and 10 like both Mathematics and Science. Then find the students who like either Mathematics or Science.

Solution:

Let A be the set of students who like Mathematics and B be the set of students who like Science.

n(A) = 18

n(B) = 16

n(A ∩ B) = 10

Use the two-set formula to find n(A ∪ B):

\( n(A \cup B) = n(A) + n(B) - n(A \cap B) \)

\( n(A \cup B) = 18 + 16 - 10 \)

\( = 24 \)

Applications of Venn Diagrams

Venn diagrams are widely used across fields because they make relationships among categories visually clear and support algebraic reasoning. Common applications include:

• Solving problems on set operations in mathematics and reasoning tests.
• Visual explanation of logical relationships in formal logic and computer science.
• Teaching set theory and helping students develop problem-solving intuition.
• Business analysis and market segmentation where overlapping customer groups must be analysed.
• Solving analogy and reasoning questions in competitive examinations.

Solved Examples on Venn Diagram

Example 1: Set A= {1, 2, 3, 4, 5} and U= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Represent A' or Ac on the Venn diagram.

Solution:

The complement A' consists of elements in U that are not in A, namely {6, 7, 8, 9, 10}. Shade the region of U outside the circle for A to represent A'.

Example 2: In a Group of people, 50 people either speak Hindi or English, 10 prefer speaking both Hindi and English, 20 prefer only English. How many people prefer speaking Hindi? Explain both by formula and by Venn diagram.

Solution:

Let H be the set of Hindi speakers and E be the set of English speakers.

n(H ∪ E) = 50

n(H ∩ E) = 10

Number who prefer only English = 20, so n(E only) = 20.

Compute total English speakers n(E):

n(E) = n(E only) + n(H ∩ E)

n(E) = 20 + 10

n(E) = 30

Use the union formula to find n(H):

\( n(H \cup E) = n(H) + n(E) - n(H \cap E) \)

\( 50 = n(H) + 30 - 10 \)

\( 50 = n(H) + 20 \)

\( n(H) = 30 \)