Venn Diagram
Discrete Mathematics

What is a Venn Diagram
• A diagram that consists of circles.
• The circles represent the elements of the set, and
the outer parts represent elements that are not
part of the set.
• A Venn Diagram is used to organize a list of data.

History of Venn Diagram
• Venn diagrams were introduced around 1880 by
John Venn. They are used to teach elementary set
theory. (*Set theory is the branch of mathematics
that studies sets, which are collections of objects)
• It is also used to illustrate simple set relationships in
probability, logic, statistics, linguistics and computer
science

History of Venn Diagram
• Venn diagrams were introduced around 1880 by
John Venn. They are used to teach elementary set
theory. (*Set theory is the branch of mathematics
that studies sets, which are collections of objects)
• It is also used to illustrate simple set relationships in
probability, logic, statistics, linguistics and computer
science

Venn diagram glossary
• Set - A collection of things. Given the versatility of Venn
diagrams, the things can really be anything. The things may
be called items, objects, members or similar terms.

Venn diagram glossary
• Union - All items in the sets.

Example - Union
• If A = {2, 5, 7} and B = {1, 2, 5, 8}. Find A U B using Venn Diagram.
Therefore, from the Venn diagram we get A U B = {1, 2, 5, 7, 8}

Example - Union
• From the adjoining figure find A union B.
Therefore, from the Venn diagram we get A U B = {0, 1, 2, 3, 5, 8, 9}

Activity - Union
1. If A = {1, 2, 3, 6, 9, 18} and B = {1, 2, 3, 4, 5}. Find A
U B using Venn Diagram.
2. If B = {1, 2, 3, 4, 5} and A = {2, 4, 6, 8}. Find A U B
using Venn Diagram.
3. Let A = {1, 2} and B = {2, 3}. Find A U B using Venn
Diagram.

Activity - Union
Find the elements of the Sets.
A=
B=
AUB=

Venn diagram glossary
• Intersection - The items that overlap in the sets. Sometimes
called a subset.

Example - Intersection
• If A = {1, 2, 3, 4, 5} and B = {1, 3, 9, 12}. Find A ∩ B using Venn
diagram.
Therefore, from the Venn diagram we get A ∩ B = {1, 3}

Example - Intersection
• From the adjoining figure find A intersection B.
Therefore, from the Venn diagram we get A ∩ B = {p, q, m}

Activity - Intersection
1. If A = {1, 2, 3, 6, 9, 18} and B = {1, 2, 3, 4, 5}. Find A
∩ B using Venn Diagram.
2. If P = {3, 6, 9, 12, 15, 18}and Q = {2, 4, 6, 8, 10, 12,
14}. Find P ∩ Q using Venn Diagram.
3. Let A = {a, e, i, o, u} and B = {z, v, x, a, o}. Find A ∩
B using Venn Diagram.

Venn diagram glossary
• Symmetric difference of two sets - Everything but the
intersection.
• Let A and B are two sets. The symmetric difference of two sets A and B is the set (A – B) ∪ (B – A) and is
denoted by A △ B.

Example - Symmetric difference
• If A = {1, 2, 3, 4, 5, 6, 7, 8} and B = {1, 3, 5, 6, 7, 8, 9}, then A – B =
{2, 4}, B – A = {9} and A △ B = {2, 4, 9}.
Therefore, the shaded part of the Venn diagram represents A △ B = {2, 4, 9}.

Example - Symmetric difference
• If A = {1, 2, 4, 7, 9} and B = {2, 3, 7, 8, 9} then A △ B = {1, 3, 4, 8}
Therefore, the shaded part of the Venn diagram represents A △ B = {1, 3, 4, 8}.

Activity - Symmetric difference
1. If P = {a, c, f, m, n} and Q = {b, c, m, n, j, k} . Find P △ Q
using Venn Diagram.
2. Suppose there are two sets with some elements.
Set A = {1, 2, 3, 4, 5}
Set B = {3, 5}
Find A △ B using Venn Diagram.

Venn diagram glossary
• Absolute complement - Everything not in the set.

Venn diagram glossary
• Relative complement - In one set but not the other.

Venn diagram glossary
• Relative complement - In one set but not the other.