Definition Types Properties and Solved Examples of Subsets
In mathematics, we define the subset as showing that all the elements of some set A are contained within some other set B. There are two types of subsets, proper subsets and normal subsets.
If all elements of set A are in another set B, then set A is said to be a subset of set B.
Subsets
What is a Subset in Maths?
Set “a” is said to be a subset of set “b” if all the elements of set “a” are present in set “b”. In alternative words, set “a” is contained within set “b”.
Example: if set “a” has {x, y} and set “b” has {x, y, z}, then “a” is the subset of “b” as elements of “a” are present in set “b”.
Subset Symbol
A subset is denoted by the symbol ⊆.
A ⊆ b;
which means set A is a subset of set b.
All Subsets of a Set
The subsets of any set contain all possible sets, including its components and the null set. Let us understand with the help of an example.
Example: find all the subsets of set a = {1,2,3,4}
Solution: given, a = {1,2,3,4}
Subsets = {}
{1}, {2}, {3}, {4},
{1,2}, {1,3}, {1,4}, {2,3} ,{2,4}, {3,4},
{1,2,3}, {2,3,4}, {1,3,4}, {1,2,4}
{1,2,3,4}.
Types of Subsets
Subsets are classified as:
Proper subset
Improper subsets
For example, if set a = {2, 4, 6}, then,
Number of subsets: {2}, {4}, {6}, {2,4}, {4,6}, {2,6}, {2,4,6} and φ or {}.
Proper subsets: {}, {2}, {4}, {6}, {2,4}, {4,6}, {2,6}
Improper subset: {2,4,6}
There is no specific formula to find the subsets. Instead, we've to list them to differentiate between proper and improper ones.
Proper Subsets
Set a is considered a proper subset of set b if set b contains a minimum of one component not present in set a.
Example: if set a has elements as {12, 24} and set b has elements as {12, 24, 36}, then set a is the proper subset of b as 36 is not present within set a.
Proper Subset Symbol
A proper subset is denoted by ⊂. We will express a proper subset for set a and set b as,
A ⊂ b
Formula
If a set has “n” components, then the number of subsets of the given set is 2n, and conjointly the number of proper subsets of the given subset is given by 2n-1.
Consider an example, if set a has the elements A = {a, b},
then the proper subset of the given set is { }, {a}, and {b}.
Here, the number of components within the set is 2.
We know that the formula to calculate the number of proper subsets is 2n – 1.
= 22 – 1
= 4 – 1
= 3
Thus, the number of proper subsets for the given set is 3 ({ }, {a}, {b}).
What is an Improper Subset?
A subset that contains all the elements of the initial set is named an improper subset of the initial set. it's denoted by ⊆.
For example:
set p ={2,4,6}
then, the subsets of p are;
{}, {2}, {4}, {6}, {2,4}, {4,6}, {2,6} and {2,4,6}.
Where, {}, {2}, {4}, {6}, {2,4}, {4,6}, {2,6} are the proper subsets and {2,4,6} is the improper subsets. Therefore, we can write {2,4,6} ⊆ p.
Power Set
The power set is said to be the collection of all the subsets. It's represented by p(A). If A is set having elements {a, b}.
Then the power set of “A” can be;
P(A) = {∅, {a}, {b}, {a, b}}.
Properties of Subsets
Some of the necessary properties of subsets are:
Every set is taken into account as a subset of the given set itself. It implies that x ⊂ x or y ⊂ y, etc.
We can say an empty set is considered a subset of each set.
X is a subset of y. It implies that x is contained in y.
If a set x could be a subset of set y, we can say that y may be a superset of x.
Solved Examples
1. How many subsets containing three elements can be formed from the set?
S = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }
Ans: Number of elements within the set = 10
Number of parts within the subset = 3
Therefore, the number of possible subsets containing 3 parts= 10c3
$=\dfrac{10!}{(10-3)!.3!}$
$=\dfrac{10 \times 9 \times 8 \times 7!}{(7)!.3 \times 2 \times 1}$
$=\dfrac{720}{6}$
$=120$
Therefore, the number of possible subsets containing 3 parts from the set S = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} is one hundred twenty.
2. Given any two real-life examples on the subset.
Ans: we can find a variety of examples of subsets in our way of life, such as:
If we consider all the books in a library as one set, then books relating to maths may be a subset.
If all the items in a grocery store form a collection, then cereals form a subset.
3. Find the number of subsets and the number of proper subsets for the given set A = {5, 6, 7, 8}.
Ans: Given: A = {5, 6, 7, 8}
The number of elements in the set is 4.
We know that,
The formula to calculate the number of subsets of a given set is 2n
= 24 = 16
The number of subsets is 16.
The formula to calculate the number of proper subsets of a given set is 2n – 1
= 24 – 1
= 16 – 1 = 15
The number of proper subsets is 15.
Conclusion
If a set has n no. of elements, then the no. of subsets of the given set is 2n, and the no. of proper subsets is 2n-1. In this article, What are a subset and its symbol, the properties of the subset, the definition of a power set and the types of subsets are explained with examples.
Practice Problems
Find the number of subsets and the number of proper subsets for the given set A = {5, 4, 2, 1, 0}.
Find the subsets for the given set A = {3, 4, 5, 6, 7}.
List of Related Articles
FAQs on Subsets in Set Theory Explained Clearly
1. What is a subset in mathematics?
A subset is a set in which every element is also an element of another set. If set A is a subset of set B, we write A ⊆ B.
- This means every element of A is contained in B.
- Example: If A = {1, 2} and B = {1, 2, 3}, then A ⊆ B.
- Subsets are a fundamental concept in set theory and discrete mathematics.
2. What is the symbol for subset?
The symbol for subset is ⊆, and for proper subset it is ⊂.
- A ⊆ B means A is a subset of B (possibly equal).
- A ⊂ B means A is a proper subset of B (not equal to B).
- The symbol ⊄ means “not a subset of.”
3. What is the difference between a subset and a proper subset?
A subset may be equal to the original set, while a proper subset must be strictly smaller than the original set.
- If A = {1,2,3} and B = {1,2,3}, then A ⊆ B but not A ⊂ B.
- If C = {1,2}, then C ⊂ B because C does not contain all elements of B.
- Every proper subset is a subset, but not every subset is proper.
4. How many subsets does a set have?
A set with n elements has 2ⁿ subsets.
- This includes the empty set and the set itself.
- Example: If A = {1,2,3}, then number of subsets = 2³ = 8.
- This formula is widely used in combinatorics and power set calculations.
5. What is the empty set and is it a subset of every set?
The empty set (∅) is a set with no elements and it is a subset of every set.
- Symbol: ∅ or { }.
- For any set A, ∅ ⊆ A.
- This is because there are no elements in ∅ that violate the subset condition.
6. What is a power set?
The power set of a set is the set of all possible subsets of that set.
- Denoted by P(A).
- If A has n elements, then |P(A)| = 2ⁿ.
- Example: If A = {1,2}, then P(A) = {∅, {1}, {2}, {1,2}}.
7. How do you find all subsets of a set?
To find all subsets of a set, list every possible combination of its elements including the empty set and the set itself.
- Step 1: Write the empty set ∅.
- Step 2: List all single-element subsets.
- Step 3: List combinations of two elements, three elements, etc.
- Step 4: Include the full set.
- Example: For {a,b}, subsets are ∅, {a}, {b}, {a,b}.
8. Can a set be a subset of itself?
Yes, every set is a subset of itself, meaning A ⊆ A is always true.
- This follows directly from the definition of subset.
- However, a set is not a proper subset of itself.
- This property is called the reflexive property of subsets.
9. What are equal sets and how are they related to subsets?
Two sets are equal if they contain exactly the same elements, meaning A ⊆ B and B ⊆ A.
- Equal sets have identical elements regardless of order.
- Example: {1,2,3} and {3,2,1} are equal sets.
- Subset relations help determine set equality.
10. What are common mistakes when working with subsets?
Common mistakes with subsets include confusing elements with subsets and miscounting total subsets.
- Remember: If 1 ∈ A, it does not mean {1} ⊆ A automatically without checking.
- Do not forget the empty set when counting subsets.
- Use the formula 2ⁿ carefully, where n is the number of elements.
- Distinguish clearly between ⊆ (subset) and ⊂ (proper subset).
