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 well-defined 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.

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.

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.

The number of subsets of a set with n elements is 2ⁿ. 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 = 2³ = 2^{number of elements of A}. Thus, the formula to find the number of subsets of a set with 'n' elements is 2ⁿ. Here are more examples:

• If A has 2 elements, it has 2² = 4 subsets.
• If A has 5 elements, it has 2⁵ = 32 subsets.
• If A has 0 elements, it has 2⁰ = 1 subset (which is the empty set Φ)

There are two subset symbols.

1. ⊆, which is read as "is a subset or equal to" (or sometimes simply as "subset of")
2. ⊂, 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.

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 2ⁿ - 1. We have already seen that the number of subsets of a set with 'n' elements is 2ⁿ. 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 2ⁿ. For example:

• The number of proper subsets of A = {1, 2, 3} is, 2³ - 1 = 7.
• The number of proper subsets of A = {a, b} is, 2² - 1 = 3.
• The number of proper subsets of empty set Φ = { } is, 2⁰ - 1 = 0.

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.

The following are the differences between proper and improper subsets. For these differences, consider a set A with 'n' elements in it..

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 2ⁿ elements as we have already seen that a set with 'n' elements has 2ⁿ 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 2² = 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 2³ = 8 elements.

Let us summarize all the formulas related to subsets. If a set A has 'n' elements:

• The number of subsets of A = 2ⁿ.
• Hence, the number of elements of the power set of A (P(A)) is 2ⁿ.
• The number of proper subsets of A is 2ⁿ - 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 2ⁿ where 'n' is the number of elements in the set.

☛Related Topics:

• Difference of Sets
• Types of Sets
• Union of Sets
• Intersection of Sets

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 2ⁿ. So the number of subsets of a set of order 3 is 2³ = 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 2ⁿ. 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 2ⁿ 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 2ⁿ subsets.