A subset is a set whose all elements are contained within another set. A subset is indicated by the symbol '⊆' and read as 'is a subset of' in set theory.
In the figure below, every element of set A belongs to set B; A is called a subset of B.
- A subset may contain some or all elements of another set.
- Every set is a subset of itself, and the empty set ∅ is a subset of all sets.
Examples
1) For a set S = {x, y, z}, its the possible subsets are
{}, {x}, {y}, {z}, {x, y}, {y, z}, {z, x} or {x, y, z}
2) If A = {odd numbers} and B = {1, 3, 5}, so
B ⊆ A, and A is a superset of B.
3) If A = {apple, banana} and set B contains {all fruits}, then
A is a subset of B.
Subsets of Real Numbers
Real numbers that can be expressed as decimal numbers fall into a variety of categories. From your daily existence, you are undoubtedly already familiar with fractions, decimals, and counting numbers. The following numbers are considered subset of real numbers:
Subsets of Integers
Integers are a set of numbers that include all the whole numbers (both positive and negative) along with zero. They do not include fractions, decimals, or numbers with a fractional component. The set of integers is typically denoted by the symbol Z.
Example: To which subsets of the real numbers does -5 belong?
-5 is a rational number and an integer.
Power Set of a Set
A set's power set consists of every subset as well as the original set and the empty set. P(A) stands for the power set of a given set A. For example, If A = {1, 2}, then P(A) = {{ }, {1}, {2}, {1, 2}}. Here we can clearly see that all the subsets of A are contained in the P(A) ,i.e., power set of A.
Number of Subsets of a Set
For any set A, number of subsets are given using the following formula
Number of Subsets = 2n
- Where n is number of elements in the set.
As power set contain all the subsets of any set, thus for a set A which has 'n' elements then P(A) has 2n elements.
Example: How many elements of power set can be formed if there are four elements in a set?
Number of elements of power set with four elements are 24 = 16.
Types of Subsets in Maths
There are two types of subsets that are:
Proper Subset
A proper subset only comprises a few members of the original set. Proper subset can never be equal to the original set. The number of elements of a proper subset is always less than the parent set.
A proper subset is denoted by ⊂,
We can express a proper subset for set A and set B as; A ⊂ B
Example: Let set A = {1, 3, 5}, then proper subsets of A are {}, {1}, {3}, {5}, {1, 3} {3, 5} {1, 5}.
Also, {1, 3, 5} is not a proper subset of A as the number of elements is not less than the number of elements of A.
Number of Proper Subset: The number of proper subsets of a set with 'n' elements is 2n - 1.
Example: A set contain 3 elements, what will be the number of proper subsets?
Number of proper subsets = 2n - 1
Here, n = 3
N = 23 - 1 = 7
Improper Subset
An "improper subset" of a set refers to the subset that is exactly the same as the original set itself. In other words, if you have a set A, the improper subset of A is A itself
Example: What will be the improper subset of set A = {1, 3, 5}?
Improper subset: {1,3,5}
Number of improper subsets: For any set with nnn elements, there is exactly one improper subset — the set itself.
Proper vs Improper Subsets
The key differences between proper subsets and improper subsets are listed in the following table:
Proper Subset | Improper Subset |
|---|---|
It contains some of the elements of a set. | It contains all the elements of a set. |
It will never equal to a give set. | It is always equal to a given set. |
The number of proper subsets of a set with 'n' elements is 2n - 1. | For a collection of 'n' elements, the number of improper subsets is always 1. |
"⊂" symbol is used only for proper subsets. | "⊆" symbol is used for improper subsets. |
Example: For a set P = {1,2} find proper and improper subset.
Solution:
Proper set is given by { }, {1} and {2}
Improper set is given by {1,2}
Subsets vs Supersets
The key differences between both subsets and supersets are listed in the following table:
| Subset | Superset |
|---|---|
| A subset is a set that contains fewer or the same elements as another set. | A superset is a set that contains all or more elements than another set. |
| The subset relationship is denoted as A ⊆ B, where A is a subset of B. | The superset relationship is denoted as A ⊇ B, where A is a superset of B. |
| {1, 2} is a subset of {1, 2, 3}. | {1, 2, 3} is a superset of {1, 2}. |
| The subset's size is less than or equal to the superset's size. | The superset's size is greater than or equal to the subset's size. |
| All elements of a subset are also elements of the superset. | A superset includes all elements of the subset and possibly more. |
| A set can have multiple subsets. | A set can have multiple supersets. |
| The empty set (∅) is a subset of every set. | The empty set (∅) is a superset of every set. |
Subset Formulas
All the formulas related to subsets are give below.
- The number of subsets of a set with n elements is 2n. This includes both proper and improper subsets.
- The number of proper subsets of a set with n elements is 2n - 1.
- The number of improper subsets of any set is always 1.
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Solved Examples
Problem 1: How many subsets in a set with 4 elements?
Solution:
A set containing 4 elements will have 24 elements in it = 16.
Problem 2: How many subsets in a set with 5 elements?
Solution:
A set containing 5 elements will have 25 elements in it = 32.
Problem 3: Calculate the number of proper subsets for the given set A (A = 5, 6, 7, 8) .
Solution:
A proper subset of a set is a subset that is not equal to the original set and the set A = {5, 6, 7, 8} contains 4 elements.
The number of proper subsets of a set with n elements is given by:
2ⁿ − 1 Substituting n = 4:
2⁴ − 1 = 16 − 1 = 15 Therefore, the number of proper subsets of set A is 15.
Problem 4: Consider the following two sets: X = {a, b, c} and Y = {a, b, c}. Is set X a proper subset of set Y?
Solution:
For X to be a valid subset of Y, it must have fewer elements than Y and exclude at least one element from Y.
X = {a, b, c}
Y = {a, b, c}
All of the components in X are likewise present in Y, and both sets include the same elements.
Therefore, X is not a proper subset of Y since it does not exclude any items and is equivalent to Y.
Problem 5: A is a subset of B. If A = {x: x is an even natural number} and B = {y : y is a natural number}.
Solution:
The statement above is correct.
- A is subset of B because every elememt of A is also in B.
- A is proper subset of B because B contains elements that A does not( eg 1, 3, 5........)
Practice Problems on Subsets in Maths
Problem 1: Given a set A = {1, 2, 3, 4}, how many subsets does A have?
Problem 2: List all the subsets of the set B = {a, b}.
Problem 3: Given the set C = {x, y, z}, how many proper subsets does C have?
Problem 4: Determine if D = {2, 3} is a subset of E = {1, 2, 3, 4}.
Problem 5: Set P = {red, blue, green} and set Q = { }. Is Q a proper subset of P?