Set Theory: Venn Diagrams And Subsets


What Is A Venn Diagram?

A Venn Diagram is a pictorial representation of the relationships between sets.

We can represent sets using Venn diagrams. In a Venn diagram, the sets are represented by shapes; usually circles or ovals. The elements of a set are labeled within the circle.




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Related Pages
More Lessons On Sets
Intersection Of Two Sets
Intersection Of Three Sets

The following diagrams show the set operations and Venn Diagrams for Complement of a Set, Disjoint Sets, Subsets, Intersection and Union of Sets. Scroll down the page for more examples and solutions.

Set Operations and Venn Diagrams

The set of all elements being considered is called the Universal Set (U) and is represented by a rectangle.

  • The complement of A, A', is the set of elements in U but not in A. A' ={x | x ∈ U and x ∉ A}
  • Sets A and B are disjoint sets if they do not share any common elements.
  • B is a proper subset of A. This means B is a subset of A, but B ≠ A.
  • The intersection of A and B is the set of elements in both set A and set B. A ∩ B = {x | x ∈ A and x ∈ B}
  • The union of A and B is the set of elements in set A or set B. A ∪ B = {x | x ∈ A or x ∈ B}
  • A ∩ ∅ = ∅
  • A ∪ ∅ = A



Set Operations And Venn Diagrams

Example:
1. Create a Venn Diagram to show the relationship among the sets.
U is the set of whole numbers from 1 to 15.
A is the set of multiples of 3.
B is the set of primes.
C is the set of odd numbers.

2. Given the following Venn Diagram determine each of the following set.
a) A ∩ B
b) A ∪ B
c) (A ∪ B)'
d) A' ∩ B
e) A ∪ B'

Example:
Given the set P is the set of even numbers between 15 and 25. Draw and label a Venn diagram to represent the set P and indicate all the elements of set P in the Venn diagram.

Solution:
List out the elements of P.
P = {16, 18, 20, 22, 24} ← ‘between’ does not include 15 and 25
Draw a circle or oval. Label it P. Put the elements in P.



Example:
Draw and label a Venn diagram to represent the set
R = {Monday, Tuesday, Wednesday}.

Solution:
Draw a circle or oval. Label it R . Put the elements in R.



Example:
Given the set Q = { x : 2x – 3 < 11, x is a positive integer }. Draw and label a Venn diagram to represent the set Q.

Solution:
Since an equation is given, we need to first solve for x.
2x – 3 < 11 ⇒ 2x < 14 ⇒ x < 7



So, Q = {1, 2, 3, 4, 5, 6}
Draw a circle or oval. Label it Q.
Put the elements in Q.



Venn Diagram Videos

What’s a Venn Diagram, and What Does Intersection and Union Mean?

Venn Diagram - Two Subsets

Learn about Venn diagrams with two subsets.

Venn Diagram - Two Subsets Using Regions

Learn about Venn diagrams with two subsets using regions.

Venn Diagrams With Three Subsets

Learn about Venn diagrams with three subsets.

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