Sets in Maths: Definition, Examples & Types - GeeksforGeeks (2025)

In mathematics, a set is simply a collection of distinct objects, called elements or members, grouped together because they share some property or characteristic. You can think of it like a "basket" where you collect items that fit a certain rule or idea.

For example:

• A set of all vowels in the English alphabet: {a, e, i, o, u}
• A set of numbers greater than 5: {6, 7, 8, 9, 10, ...}

Definition of Sets

A setA is written as:

A = {x ∣ propertyofx}

This meansAis the set of allxsuch thatxsatisfies a certain property.

Key characteristics of sets are:

• Well-defined: The contents of the set are clearly specified and identifiable.
  • Example: The set of natural numbers less than 5 is {1, 2, 3, 4}.

• Distinct Elements: A set cannot have duplicate elements.
  • Example: {1, 2, 2, 3} is the same as {1, 2, 3}.

• Order of Elements: The order of elements does not matter.
  • Example: {1, 2, 3} is the same as {3, 2, 1}.

Examples of Sets

Some of the common examples of sets are:

Finite Sets

• A set of vowels in the English alphabet: A = {a, e, i, o, u}
• A set of natural numbers less than 6: B = {1, 2, 3, 4, 5}
• A set of primary colors: C = {red,blue,yellow}

Infinite Sets

• A set of natural numbers: N = {1, 2, 3, 4, . . .}
• A set of integers: Z = {. . . , -3, -2, -1, 0, 1, 2, 3, . . . }
• A set of real numbers greater than 0: R⁺ = {x∣x > 0}

Empty (Null) Set

• A set of months with 32 days: ∅ = {}
• A set of natural numbers less than 1: ∅ = {}

Real-World Examples of Set

• A set of planets in the Solar System: P = {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune}

Types of Sets

Sets can be classified based on their properties and characteristics. Some common types of sets are:

• Singleton Set
• Empty Set
• Finite Set
• Infinite Set
• Equal Set
• Equivalent Set

Operations on Sets

Operations on sets are rules that combine, compare, or manipulate sets to create new sets. Some of these operations are:

• Union: A ∪ B = {x ∣ x∈ Aorx ∈ B}
• Intersection: A ∩ B = {x ∣ x ∈ Aandx ∈ B}
• Difference: A − B = {x ∣ x ∈ Aandx ∉ B}
• Complement: If U is the universal set, the complement of A is: A^c = U − A.

Solved Examples on Sets

Question 1: Given the set A = {1, 2, 3, 4, 5}, identify the type of set.

Solution:

The set A = {1, 2, 3, 4 ,5} is a Finite Set because it contains a definite number of elements (5 elements in this case).

Question 2: Let A = {1, 2, 3} and B = {3, 4, 5}. Find the union of sets A and B.

Solution:

The union of two sets A and B is the set of all elements that are in A, B, or both.

A ∪ B = {1, 2, 3, 4, 5} (Notice the duplicate element 3 is only counted once).

Question 3: Let A = {2, 4, 6, 8} and B = {1, 2, 3, 4}. Find the intersection of sets A and B.

Solution:

The intersection of two sets A and B is the set of all elements that are common to both sets.

A ∩ B = {2, 4}

Question 4: Let A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7}. Find the difference A − B.

Solution:

The difference A − B consists of elements that are in A but not in B.

A − B = {1, 2}

Question 5: Let U = {1, 2, 3, 4, 5, 6, 7} be the universal set, and A = {2, 4, 6}. Find the complement of A (denoted as A^c).

Solution:

The complement of set A, denoted A^c, consists of all the elements in the universal set U that are not in A.

A^c = U − A = {1, 3, 5, 7}

Unsolved Practice Questions

Question 1: Given the set B = {a, e, i, o, u}, identify the type of set.

Question 2: Let X = {1, 2, 3, 4} and Y = {3, 4, 5, 6}. Find the union of sets X and Y.

Question 3: Let M = {2, 5, 7, 10} and N = {1, 5, 9, 10}. Find the intersection of sets M and N.

Question 4: Let P = {1, 3, 5, 7, 9} and Q = {2, 4, 6, 8, 10}. Find the difference P − Q.

Question 5: Let U = {2, 4, 6, 8, 10, 12, 14} be the universal set, and S = {6, 8, 12}. Find the complement of S (denoted as S^c).

Read More,

• Set Theory
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• Power Set
• Set Symbols
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