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105 LearnersLast updated on October 23, 2025
Subset
A subset is a fundamental concept in set theory. A set is a collection of different elements, such as numbers, letters, or objects. For example, a set can be written as {a, b, c, d}. A set A is a subset of set B if every element of set A is also an element in set B. For example, if set A = {10, 20, 30, 40} and set B = {10, 20, 30, 40, 50, 60}, then set A is a subset of set B. In this article, we will learn about subsets, types of subsets, and the symbols used to represent a subset.
What is a Subset?
A subset is a set whose elements are also a part of another set. For instance, if all elements in the set A are a part of set B, then set A is a subset of set B. A subset is represented using the symbol ⊆, in set theory. For example, if A = {a, b, c} and B = {a, b, c, d, e, f}, then A is a subset of B.
To find the total number of subsets of a set, we use the formula: 2n, where n is the number of elements in the set. For example, if A = {2, 4, 6}, then the number of subsets of a set A 23 = 8.
Difference Between Subset and Superset
The subset and superset are types of sets. If A is a subset of B, then B is a superset of A. In this section, we will discuss the difference between a subset and a superset.
| Subset |
Superset |
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A set is said to be a subset only if all its elements are also elements of the second set |
A set is a superset of another set if it contains all the elements or additional elements than another set |
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If A is a subset of B, it can be represented as A ⊆ B |
If set A is the superset of set B, it can be represented as A ⊇B |
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For example, if A = {p, q, r} and B = {p, q, r, s, t}, Then A is a subset of B |
For example, if A = {p, q, r} and B = {p, q, r, s, t}, Then B is a superset of A |
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The subset size is smaller than or equal to the superset. |
The superset has all the elements of the subset. |
What are the Types of subsets?
Subsets are classified into different types based on the number of elements they have. The main types of subsets are:
- Proper Subset
- Improper Subset
- Singleton Subset
Proper Subset: A proper subset includes all sets except the set itself. For example, if A = {3, 6, 9}, then its proper subsets are {}, {3}, {6}, {9}, {3, 6}, {3, 9}, {6, 9}, but the set {3, 6, 9} is not a proper subset. If A is a proper subset of B, then it can be represented as A ⊂ B, where A ≠ B, which means elements in set A are a part of set B, but A does not have all elements of B.
Improper Subset: The improper subset of a set is the set itself. For example, for the set {3, 6, 9}, the improper set is {3, 6, 9}. If A ⊆ B and A = B, then A is a subset of B, but not a proper subset.
Singleton Subset: The subset of a set that contains only one element of the set is a singleton subset. For example, A = {5, 6, 7}, then the singleton subset will be {5}, {6}, {7}.
What is the Symbol of Subsets?
In set theory, to represent a subset, we use symbols like ⊆ and ⊂. To represent the proper subset, we use the symbol ⊂, and to represent the improper subset, we use the symbol ⊆. The symbol ⊆ is read as a subset or equal to, and ⊂ is read as a subset of.
What is a Proper Subset?
A proper subset of a set is a subset that contains some elements, but not all, elements of the original set. It does not include the set itself. For example, if A = {2, 4, 6}, then its proper subset are: {}, {2}, {4}, {6}, {2, 4}, {4, 6}, and {2, 6}, but {2, 4, 6} is a subset but not a proper subset. A proper subset can be represented using the symbol ⊂, where A ⊂ B and A ≠ B. The number of proper subsets of a set can be calculated using the formula: 2n - 1, where n is the number of elements.
What is a Subset of Real Numbers?
The numbers, including positive integers, negative integers, fractions, and irrational numbers, are the real numbers. The subset of real numbers includes:
- Rational Numbers: The numbers in the form p/q, where p and q are integers and q ≠ 0, and include terminating and repeating decimals.
- Integers: The numbers include all the whole numbers and integers, including zero.
- Whole Numbers: The numbers include all positive numbers and zero.
- Natural Numbers: The numbers include all positive integers.
What is a Subset of Integers?
Integers include positive and negative numbers and zero. It doesn't include the fractions and decimals. To represent the set of integers, we use the symbol Z.
What is a Power Set of a Set?
The power set of a set includes all the possible subsets of the set, including the empty set and the set itself. The power set of a set is represented by p(A), where A is the original set.
For example, if A = {5, 10}, then the power set of A is denoted as p(A),
p(A) = {{}, {5}, {10}, {5, 10}}
Common Mistakes and How to Avoid Them in Subset
In mathematics, students often make mistakes when learning about subsets. Here are some common mistakes and the tips to avoid them in the subset.
Mistake 1
Confusing a subset with a proper subset
Students often mix up a subset with a proper subset. For instance, students assume that if A = {1, 2} and B = {1, 2}, then A ⊂ B, which is wrong. To avoid this error, students should remember that ⊆ means A = B allows equality, but ⊂ does not.
Mistake 2
Not including the empty set as a subset
Students sometimes forget to include the empty set when listing the subsets, thinking that subsets must have all elements. For example, students sometimes assume the subset of A = {1, 2} is {1}, {2}, {1, 2} instead {}, {1}, {2}, {1, 2}. To avoid the error, students should always include the empty set when listing the subset.
Mistake 3
Miscalculating the number of subsets
One of the common errors students make is miscalculating the number of subsets. They mistakenly assume that the number of subsets of a set with n elements is n, but they are wrong; the correct number is 2n. To avoid this error, always remember the formula to find the number of subsets. The number of subsets of a set with n elements is 2n.
Mistake 4
Confusing a subset with a superset
Students mix up the subset with the superset, for example, if A = {1} and B = {1, 2} students assume that {1, 2} ⊆ {1} instead of {1} ⊆ {1, 2}. So always check the direction of ⊆ to know whether the set is a subset or a superset.
Mistake 5
Misinterpreting set notation
Misinterpretation of set notation is common among students, especially with the subset (⊆) and element notation (∈). To avoid confusion, always remember that the symbol ⊆ is used to compare two sets, to indicate that one set is a subset of another, whereas ∈ is used to represent that an element is a part of the set.
Real-World Applications of Subset
In set theory, a subset is a fundamental concept and is used in different fields such as computer science, biology, economics, etc. In this section, we will learn a few real-world applications of subsets.
- In online shopping platforms, subsets are used to display products based on selected criteria. For example, if you are searching for a dress in white color, under $15, then the website will show only items based on the criteria.
- In event planning and scheduling, subsets are used to represent particular combinations.
- In biology, to model gene regulatory networks, we use subsets.
- In economics, we use subsets to divide the market into segments based on customer characteristics.
Solved Examples on Subset
Problem 1
List all subsets of A = {1, 2}
The subset of A is {}, {1}, {2}, {1, 2}
Explanation
The number of subsets of A = 2n = 22 = 4
The subset of A includes the empty set, the single set, and the set itself.
So, the subsets are {}, {1}, {2}, {1, 2}
Problem 2
If B = {x, y, z}, find how many subsets B has.
The number of subsets of B is 8
Explanation
The number of subsets of a set = 2n
Here, n = 3
So, the number of subsets of B = 23 = 8
Problem 3
Is {5, 6} ⊂ {5, 6, 7}?
Yes, {5, 6} ⊂ {5, 6, 7}
Explanation
Let’s consider A = {5, 6} and B ={5, 6, 7}
The elements in set A are also in set B, so set A is a proper subset of set B. So, {5, 6} ⊂ {5, 6, 7}
Problem 4
List all the subsets of {4, 8, 12}?
The subsets of {4, 8, 12} are {}, {4}, {8}, {12}, {4, 8}, {4, 12}, {8, 12}, {4, 8, 12}
Explanation
The subsets of a set include the empty set and all the combinations of the set.
Problem 5
List all the proper and improper subsets of C = {a, b}
For C, the proper subsets are {}, {a}, {b}, the improper subset is {a, b}
Explanation
A proper subset of a set has all the elements of the set, and an improper subset contains all the elements of the set.
FAQs on Subset
1.What is a subset?
2.What are the subsets of a = {1, 2, 3}?
3.What does a ⊂ b mean?
4.What is the difference between ⊆ and ⊂?
5.What is the difference between proper and improper subsets?
Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
