Proper Subset is a set that contains some, but not all, of the members of another set. Formally, A is considered a proper subset of B if every member of set A is also an element of set B, but there is at least one element in B that is not in A. For any proper subset, its cardinality (the number of items) is always fewer than the cardinality of the set from which it is generated.
This article covers proper subsets in detail, including their definition, examples, and symbols. In addition to that, we will also discuss improper subsets and the key differences between proper and improper subsets in this article.
Table of Content
What is Subset?
A set ‘A’ is said to be subset of Set B is all the elements of Set A is included in the set B. For Example if we take a set of natural number ‘N’ and a subset of whole numbers ‘W’ then we can say that N is a subset of W as all the elements of set N are included in the set W.
Learn More, Subset in Maths
What is a Proper Subset?
A proper subset is a subset of a bigger set that has fewer members while still including unique elements from the original set. In other words, a proper subset is constructed by picking a subset of items from a larger set while removing at least one member from the larger set. If set A is a proper subset of set B, it means that all of A’s items are also present in B, but B has at least one member that is not in A. The sign ‘⊂’ denotes a proper subset.
Definition of Proper Subset
For two sets A and B, A is said to be a proper subset of B i.e., A ⊂ B; if and only if for every element x in A, x is also in B, and there exists at least one element y in B such that y is not in A.
Which can be written in mathematical notation as follows:
∀x (x ∈ A → x ∈ B) ∧ ∃y (y ∈ B ∧ y ∉ A)
Example of Proper Subset
The example of Proper Subset is listed below:
- Natural Numbers: Set of even natural numbers (E) is a proper subset of the set of all natural numbers (N). E = {2, 4, 6, 8, …} and N = {1, 2, 3, 4, 5, 6, …}.
- Alphabet Letters: The set of vowels (V) is a proper subset of the set of all English alphabet letters (A) as V = {a, e, i, o, u} and A = {a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z}.
- Geometric Shapes: The set of quadrilaterals (Q) is a proper subset of the set of polygons (P). Q = {square, rectangle, rhombus, trapezoid, …} and P = {triangle, pentagon, hexagon, octagon, …}.
- Musical Instruments: The set of string instruments (S) is a proper subset of the set of all musical instruments (M) as S = {guitar, violin, cello, bass, …} and M = {piano, flute, trumpet, drums, guitar, violin, …}.
Proper Subset Symbol
In mathematics, the proper subset symbol is expressed as A⊂B, signifying that set A is a subset of but not equal to set B. This means that every element in A is likewise present in B, but B has at least one more element.
For example, let’s consider two sets A = {1,2} and B= {1,2,3,4}. Here, we can use the subset symbol to represent the relationship between set A and B i.e., A ⊂ B. As every element in A i.e., 1 and 2, is likewise present in B, but B has additional elements i.e., 3 and 4; that are not present in A.
Learn more about Set Symbols.
Proper Subset Formula
If a set has “n” items, the number of subsets for the supplied set is 2n, and the number of appropriate subsets of the provided subset is computed using the formula 2n – 1.
What is Improper Subset?
An improper subset is a subset of a set that includes all the elements of the original set, along with the possibility of being equal to the original set itself. In other words, if set A is considered an improper subset of set B, it means that every element of set A is also an element of set B, and set A may or may not be equal to set B.
Mathematically, if A is an improper subset of B, it is represented as:
A ⊆ B
Where,
- “⊆” denotes the subset relationship,
- A is the subset, and
- B is the superset.
Proper and Improper Subset
There are various difference between proper and improper subsets, some of those key differences are listed in the following table:
|
Proper Subset |
Improper Subset |
|---|---|
| A subset that has fewer items than the original set and contains at least one element less than the original set. | A subset that is equal to or contains all of the items of the original set. |
| There are fewer elements than in the original set. | The original set has the same or an equal number of items. |
| A valid subset of B is expressed by the symbol A ⊂ B, which means that A is included inside B but not equal to B. | An improper subset of B is indicated as A ⊆ B, which means that A is included within B and can be equal to B. |
Subset vs Proper Subset
As we know, If every element of set A is also an element of set B, then set A is called a subset of set B and proper subset is one of the type of subsets. Here, we have discussed all the key differences between Subset and Proper Subset:
| Subset | Proper Subset |
|---|---|
| A is a subset of B if all of its elements are in B and A is equal to B (A ⊆ B). | If all of A’s elements are in B, but A is not equal to B (A ⊂ B), A is a legitimate subset of B. |
| Subset can be equal to the set of which it is a subset. (A and B can be equal.) | A proper subset is always strictly smaller than the set from which it is derived. (A and B are never equal.) |
| If A = {1, 2} and B = {1, 2, 3}, then A is a subset of B (A ⊆ B). | If A = {1, 2} and B = {1, 2, 3}, then A is a proper subset of B (A ⊂ B). |
| The empty set (∅) is a subset of all sets, including itself. | The empty set (∅) is a valid subset of all non-empty sets. |
Note: All proper subsets are subsets but not all subsets are proper subsets.
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Solved Example of Proper Subset
Example 1: A subsets B. If A = {x: x is an even natural number} and B = {y: y is a natural number}.
Solution:
The statement above is correct. Because every even natural number is also a natural number, A is a subset of B. As a result, the set A, which is made up of even natural numbers, is a subset of the set B, which is made up of all natural numbers.
Example 2: The number of proper subsets for the given set A (A = 5, 6, 7, 8) is 15.
Solution:
The above statement is also incorrect. A valid subset of a set A is one that is not the same as A itself. Because the set A = 5, 6, 7, 8 includes four elements, it has 24 – 1 = 15 appropriate subsets.
Example 3: Consider the following two sets: A = {1, 2} and B = {1, 2, 3, 4}. Is set A a proper subset of set B?
Solution:
To test if A is a valid subset of B, we look at whether all of A’s items are in B and B includes at least one element that is not in A.
A = {1, 2} B = {1, 2, 3, 4}
All of A’s elements (1 and 2) are present in B, and B additionally has items (3 and 4) that are not found in A. As a result, A is a proper subset of B, which is denoted by A ⊂ B.
Example 4: Consider the following two sets: X = {a, b, c} and Y = {a, b, c}. Is set X a proper subset or improper 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. However, X does not eliminate any of the elements that Y possesses. X is not a suitable subset of Y since it does not exclude any items and is equivalent to Y. It is an incorrect subset, which may alternatively be written as X ⊆ Y.
The principles of proper and improper subsets are shown in both cases by comparing the members of the subsets to those of the larger sets.
Example 5: Determine the number of subsets and proper subsets for the given set P = {1, 2, 3, 4, 5}.
Solution:
P = {1, 2, 3, 4, 5} is given.
The set has five components.
2n is the formula for calculating the number of subsets of a given set.
Thus, Number of Subset = 25 = 32
2n – 1 is the formula for calculating the number of proper subsets of a given set.
Thus, Number of Proper Subset = 25 – 1
⇒ Number of Proper Subset = 32 – 1 = 31
Therefore, the number of proper subset for P = {1, 2, 3, 4, 5} is 31.
Practice Questions on Subset
Q1. Set A = {1, 2, 3, 4} and set B = {2, 3}. Determine whether B is a proper subset of A.
Q2. Set X = {a, b, c, d} and set Y = {a, b, c, d}. Are X and Y proper subsets of each other?
Q3. Set P = {red, blue, green} and set Q = {}. Is Q a proper subset of P?
Q4. Set M = {1, 2, 3, 4, 5} and set N = {1, 2, 3, 4, 5}. Are M and N proper subsets of each other?
Q5. Set R = {apple, banana, orange, grape} and set S = {banana, orange}. Is S a proper subset of R?
FAQs on Proper Subset
1. What do you Mean by Proper Subset?
A proper subset is a subset of a set that contains some but not all elements of the original set. It is denoted as A ⊂ B, where A is a proper subset of B.
2. Is an Empty set a Proper Subset?
No, an empty set (∅) is not a proper subset of any set as It contains no elements .
3. Can we find Subsets of Infinite Set as Well?
Yes, you can find subsets of infinite sets. Infinite sets, like the set of natural numbers (N), can have subsets, such as the set of even natural numbers (E) or the set of prime numbers (P).
4. What is the Difference Between a Proper and an Improper Subset?
A proper subset is a subset that contains some but not all elements of the original set. An improper subset contains all elements of the original set and is the same as the original set itself.
5. Define Proper Subset.
A proper subset is a set A that contains some, but not all, elements of set B.
6. What is a Proper Subset Symbol?
The symbol for a proper subset is “⊂”. So, if set A is a proper subset of set B, it would be denoted as A⊂B.
7. What is the Difference between ⊆ and ⊂?
The symbol ⊆ represents “is a subset of or equal to,” meaning that a set A ⊆ B can be equal to set B or a proper subset. On the other hand, ⊂ represents “is a proper subset of,” indicating that a set A ⊂ B is always a proper subset, not equal to B itself.
8. What is an Example of a Proper Subset?
Consider two sets, A = {1, 2, 3} and B = {1, 2, 3, 4}. In this case, set A is a proper subset of set B. All elements in set A (1, 2, and 3) are also present in set B, but B contains an additional element (4) not found in A.
Therefore, A is a proper subset of B, denoted as A ⊂ B.
9. What are the Proper Subset of A = {1, 2, 3}?
The proper subsets of A = {1, 2, 3} are:
- {} (the empty set)
- {1}
- {2}
- {3}
- {1, 2}
- {1, 3}
- {2, 3}
Each of these subsets contains some, but not all, elements of the original set A.
10. How many Subsets does the Set A = {1, 2, 3, 4, 5} can have?
The set A = {1, 2, 3, 4, 5} have 25 = 32 subsets, including the empty set and the set itself.
11. Is Null Set a Proper Subset?
No, the null set (∅) is not considered a proper subset. It is a subset of every set, but it is not a proper subset of any set because it does not contain “some” but “no” elements of the original set.