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.
Proper Subset Formula
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
An improper subset contains includes both the null set and each member of the initial set. Improper subset is equal to the original set. This is represented by the symbol ⊆.
Example: What will be the improper subset of set A = {1, 3, 5}?
Improper subset: {}, {1}, {3}, {5}, {1,3}, {1,5}, {3,5} and {1,3,5}
Improper Subset Formula
For a collection of 'n' elements, the number of improper subsets is always 1. In other words, the number of improper subsets of a set is independent of the number of its elements.
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 on Subsets in Maths
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 valid subset of a set A is one that is not the same as A itself.
As the set A = 5, 6, 7, 8 includes four elements, it has 24 – 1 = 15 proper subsets.
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?