Subset
If all elements of set 1 are present in set 2 then we say that set 1 is a subset of set 2. We know that a set is a welldefined collection of numbers, alphabets, objects, or any items. If set 1 = {A,B,C} and set 2 = {A,B,C,D,E,F} we can say that set 1 is a subset of set 2 since all the elements in set 1 are present in set 2.
Let us learn about the subsets along with their types (proper subset and improper subset) with many examples.
What is a Subset?
A subset is a part of a given set (another set or the same set). The set notation to represent a set A as a subset of set B is written as A ⊆ B.
Subset Meaning
If all elements of set A are in another set B, then set A is said to be a subset of set B. In this case, we say
 A is a subset of B (or)
 B is a superset of A
For example, A is the set of natural numbers, and B is the set of all whole numbers, then A is a subset of B because all natural numbers are present in the set of whole numbers). We can understand it this way:
 A = Set of natural numbers = {1, 2, 3, ....}
 B = Set of whole numbers = {0, 1, 2, 3, ...}
 Since every element of A is in B, A ⊆ B.
Subset Examples
For a set A to be a subset of a set B, the only condition is every element of A should be present in B. Based upon this, here are a few subsets examples.
 A = {1, 2, 3} is a subset of B = {1, 2, 3, 4, 10}
 A = {p, q, r} is a subset of B = set of all alphabets
 A = set of all even numbers is a subset of B = set of all integers
Note that every set is a subset of itself and also the empty set (Φ) is also a subset of every set.
Number of Subsets of a Set
The number of subsets of a set with n elements is 2^{n}. For example, if A = {1, 2, 3}, then the number of elements of A = 3. The subsets of A are { }, {1}, {2}, {3}, {1, 2}, {2, 3}, {3, 1}, and {1, 2, 3}. So A has totally 8 subsets and 8 = 2^{3} = 2^{number of elements of A}. Thus, the formula to find the number of subsets of a set with 'n' elements is 2^{n}. Here are more examples:
 If A has 2 elements, it has 2^{2} = 4 subsets.
 If A has 5 elements, it has 2^{5} = 32 subsets.
 If A has 0 elements, it has 2^{0} = 1 subset (which is the empty set Φ)
Subset Symbol
There are two subset symbols.
 ⊆, which is read as "is a subset or equal to" (or sometimes simply as "subset of")
 ⊂, which is read as "is a subset of" (means strictly subset of but NOT equal to)
In each of the above examples, we can write A ⊂ B (or) A ⊆ B. But there is a small difference between these two symbols and the usage of each symbol depends upon the type of subset. There are 2 types of subsets:
 Proper subset (The symbol used for this is ⊂)
 Improper subset (The symbol used for this is ⊆)
Let us learn more about the proper and improper subsets in the upcoming sections.
Proper Subset
A proper subset is any subset of the set except itself. We know that every set is a subset of itself but it is NOT a proper subset of itself. For example, if A = {1, 2, 3}, then its proper subsets are {}, {1}, {2}, {3}, {1, 2}, {2, 3}, and {3, 1}, but the set itself {1, 2, 3} is NOT a proper subset of A.
Proper Subset Symbol
The proper subset symbol is ⊂. i.e., if A is a proper subset of B, then:
 A ⊂ B and
 A ≠ B
Proper Subset Formula
The number of proper subsets of a set with 'n' elements is 2^{n}  1. We have already seen that the number of subsets of a set with 'n' elements is 2^{n}. Since all the subsets of a set except the set itself are the proper subsets of the set, the number of proper subsets is obtained by subtracting 1 from 2^{n}. For example:
 The number of proper subsets of A = {1, 2, 3} is, 2^{3}  1 = 7.
 The number of proper subsets of A = {a, b} is, 2^{2}  1 = 3.
 The number of proper subsets of empty set Φ = { } is, 2^{0}  1 = 0.
Improper Subset
An improper subset is a subset of the set which is NOT a proper subset. i.e., every set A has only one improper subset which is the set A itself. Here are some examples of improper subsets.
 {1, 2, 3} is the only improper subset of {1, 2, 3}
 {a, b} is the only improper subset of {a, b}
For writing the improper subsets, we usually use the symbol ⊆. i.e., if A is an improper subset of B, then A ⊆ B (A = B here). Of course, since "⊆" means "subset or equal to", this symbol can be used for the proper subsets as well. But the "strictly subset" symbol ⊂ cannot be used for writing improper subsets.
Improper Subset Formula
The number of improper subsets of a set with 'n' elements is always 1. i.e., the number of improper subsets of a set does NOT depend upon the number of elements of the set.
Differences Between Proper and Improper Subsets
The following are the differences between proper and improper subsets. For these differences, consider a set A with 'n' elements in it..
Proper Subset  Improper Subset 

It contains only a few (or no) elements of set A.  It contains all elements of set A. 
It is never equal to set A.  It is always equal to set A. 
The number of proper subsets of A is 2^{n}  1.  The number of improper subsets of A is just 1 (which is A itself). 
"⊂" should be used only for proper subsets.  "⊆" symbol can be used for writing both proper and improper subsets. 
Power Set of a Set
The power set of a set is a set of all the subsets (along with the empty set and the original set). The power set of a set A is denoted by P(A). If A has 'n' elements then P(A) has 2^{n} elements as we have already seen that a set with 'n' elements has 2^{n} subsets. Here are some examples of power sets.
 If A = {1, 2}, then P(A) = { { }, {1}, {2}, {1, 2} }
Observe that A has 2 elements and P(A) has 2^{2} = 4 elements.  If A = {a, b, c}, then P(A) = { { }, {a}, {b}, {c}, {a, b}, {b, c}, {c, a}, {a, b, c} }
Observe that A has 3 elements and P(A) has 2^{3} = 8 elements.
Subset Formulas
Let us summarize all the formulas related to subsets. If a set A has 'n' elements:
 The number of subsets of A = 2^{n}.
 Hence, the number of elements of the power set of A (P(A)) is 2^{n}.
 The number of proper subsets of A is 2^{n } 1.
 The number of improper subsets of A is 1.
Important Notes on Subsets:
 An empty set is always a subset of any given set.
 The original set is a subset of its own.
 A proper subset can be a set with all the combinations of elements except the original set.
 The number of a set is 2^{n} where 'n' is the number of elements in the set.
Related Topics:
Solved Examples on Subsets

Example 1: Determine whether A is a subset of B if A = {1, 3, 5}, and B = {1, 3, 5, 7, 9}.
Solution:
Since every element of A is present in B, A is a subset of B.
Thus, A ⊆ B.
Answer: A is a subset of B.

Example 2: If X = {x, y, z}, then how many proper subsets does X have? List All of them.
Solution:
The number of elements of X is, n = 3.
So the number of proper subsets of X is = 2^{3}  1 = 8  1 = 7.
A proper subset of X can be any subset except itself. So the proper subsets of X are, { }, {x}, {y}, {z}, {x, y}, {y, z}, and {z, x}.
Answer: X has 7 proper subsets namely { }, {x}, {y}, {z}, {x, y}, {y, z}, and {z, x}.

Example 3: Find the number of subsets, number of proper subsets, and the number of improper subsets of the set A = {a, b, c, d, e}.
Solution:
The number of elements of A is, n = 5. Thus,
 The number of subsets of A = 2^{5} = 32.
 The number of proper subsets of A = 2^{5}  1 = 31.
 The number of improper subsets = 1.
Answer: The number of subsets, number of proper subsets, and the number of improper subsets are 32, 31, and 1 respectively.
FAQs on Subsets
What is a Subset Definition?
The subset of a set A is a set that contains some elements of A or all elements of A. For example, the subsets of a set {p, q, r} is { }, {p}, {q}, {r}, {p, q}, {q, r}, {p, r}, and {p, q, r}.
How to Write the Subset of a Set?
We write the subset of a set using the subset sign ⊆. If A is a subset of B, we write A ⊆ B. A subset of a set can be a set with a few or no or all elements of the given set.
What is the Number of Subsets of a Set of Order 3?
If 'n' is the number of elements of a set A, then the number of subsets of A is 2^{n}. So the number of subsets of a set of order 3 is 2^{3} = 8.
What is the Difference Between Proper and Improper Subsets?
For any set A, any of its subsets except A itself is a proper subset. On the other hand, a subset that is NOT proper is called the improper subset. So a set A has only 1 improper subset (which is itself).
Is Phi a Proper Subset?
Phi (Φ) represents an empty set. Since it is present inside any set, it is a subset of any set. Further, it is a proper subset of any set except for Φ itself. i.e.,
 Φ is a proper subset of any set A where A is NOT equal to Φ.
 Φ is NOT a proper subset of Φ itself.
Is an Empty Set a Proper Subset or Improper Subset?
A proper subset of a set shouldn't be equal to the set itself. Thus, the empty set (Φ) is a proper subset of every set A as long as A is NOT equal to Φ.
What is the Number of Subsets of a Set Containing n Elements?
If a set A has 'n' elements, then the number of subsets of A is 2^{n}. If we write all the subsets in a set, then it is called the power set and hence the power set of a set with n elements also has 2^{n} elements in it.
What is the Difference Between a Subset and a Proper Subset?
A subset of a set A can be a set with no elements (empty set), or can be made with a few elements of A (or) can be made with all elements of A. But the proper subset cannot be made with all elements of A. In simple words, a subset of A that is NOT equal to A is a proper subset of A. For example, any subset of a set {1, 2, 3} is a proper subset of {1, 2, 3} except itself.
What is the Difference Between a Subset and a Superset?
Subset and superset in math are related to each other. If A is a subset of B then B is said to be the superset of A.
What are the Properties of Subsets?
Here are the properties of subsets.
 Every set is a subset of itself by its definition.
 The empty set is a subset of every set as empty set has no elements.
 A set with n elements has 2^{n} subsets.
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