Mathematics has a theory of sets; school teachers and private math tutors have been delivering lectures on sets, their types, symbols, properties, operations performed, and related topics for years. Set is the well-defined collection of elements, where elements could be anything such as a group of variables, constants, natural numbers, integers, whole numbers, etc., grouped in the curly braces, {}. The sets are named after or represented by capital English alphabets. Subsets are also a part of sets. Here we will discuss subset definition and its properties and types with examples.
By definition, a subset is the part of another given set. In other words, suppose you have two sets: Set X and Set Y, and if all the elements of Set X are also present inside the Set Y, then Set X is the subset of Set Y. in this case, you can say
or
Mathematically or symbolically, subsets are represented by ⊆.
Write X ⊆ Y to represent that Set X is the subset of Set Y.
If set Q has {B, C, D, E, F} and set R has {A, B, C, D, E, F, G}, then Q is the subset of R because all the elements or objects of Q are also present in the set R.
In mathematical form,
Q ⊆ R means Set Q is the subset of set R
Find all the subsets of set B = {0, 2, 4, 6}
Solution: Following are the subsets of set B
Subsets =
{}
{0}, {2}, {4}, {6},
{0,2}, {0,4}, {0,6}, {2,4}, {2,6}, {4,6},
{0,2,4}, {2,4,6}, {0,4,6}, {0,2,6}
{0,2,4,6}
Subsets are further classified into two parts:
Let us learn about types of subsets with example questions and find out the difference.
If a set contains only a few or no elements of another set, then it is the proper subset of the given set. For instance, set M is the proper subset of set N if set N contains at least one or more elements which are not present in set M.
The proper subset symbol is ⊂. We can express the proper subset for set M and set N as;
Another formula to calculate the number of proper subsets of a set is 2n – 1. Here, ‘n’ is the number of elements in a set.
Solution:
Given that,
A = {x, y}
Number of elements in set A = 2
Using proper subsets’ formula: 2n – 1
= 22 – 1
= 4 – 1
= 3
Thus, the number of proper subsets of the set A = 3.
Proper Subsets = ({}, {a}, {b})
If a set contains all elements of another set, it is called an improper subset of the original set. For instance, set M would be the improper subset of set N because it contains all the elements of another set.
Symbolically, improper subsets are denoted by ⊆. We can express the improper subset for set M and set N as:
A set with the collection of all the subsets is known as Power Set. The capital alphabet ‘P’ represents the power set. Therefore, the power set of set B would be written as P(B).
If B has n elements, then P(B) has 2n, which means power sets will have certain elements.
If set B has {0, 1} elements than its power set will be:
2n = 22 = 4
P(B) = {{}. {0}, {1}, {0, 1}}
If a set contains all the objects of elements of other given sets, it is known as a universal set. Mostly, the universal set is represented by the capital English Alphabet U.
If C = {2, 4, 6}, A = {1, 2, 3}, T= {6, 8, 9}
Then U = {1, 2, 3, 4, 6, 8, 9}
Hence, C ⊆ U, A ⊆ U, T ⊆ U
The following table is designed to help you understand the differences between proper and improper subsets:
Proper Subset | Improper Subset |
Contains only a few elements of set D. | Contains all the elements of set D |
Never equals to set D. | Always equal to set A. |
2n – 1 is used to calculate the number of subsets of D | The set itself is the proper subset of set D, which is 1. |
Symbolically, proper subsets are represented as “⊂” | Symbolically, improper subsets are represented as “⊆” |
Note: In the above table, mentioned D is the set with n number of elements.
Solution:
Since every element of set P is present in set O then, P is the subset of O
Thus, P ⊆ O
Hence, P is the subset of O.
Solution:
Given that,
⇒ A = {a, e, i, o, u}
Number of elements in the set A = 5
Formula to calculate the number of subsets = 2n
⇒ 25 = 32
Formula to calculate the number of proper subsets = 2n-1
⇒ 25 – 1 = 32 – 1 = 31
Hence,
Number of subsets = 32
Number of proper subsets = 31
Answer:
Set A has a single element, and Set B has four elements. Where set A represents You and set B represents Your Family. By definition, each element of a subset should be included in another set, and as ‘you’ is a part of your family, A ⊂ B. A is the subset of B.
Solution:
Given set B = {2, 4, 6}
Subsets:
{}, {2}, {4}, {6}, {2,4}, {4,6}, {2,6}, {2,4,6}
Here,
⇒ Proper subsets = {}, {2}, {4}, {6}, {2,4}, {4,6}, and {2,6}
⇒ Improper subset= {2,4,6}
Answer:
⇒ With 2 elements, 4 subsets of a set will exist.
⇒ With 3 elements, 8 subsets of a set will exist.
⇒ With 1 element, 2 subsets of a set will exist.
Answer:
Subsets are classified into Proper subset and Improper subset.
Answer:
Yes, a set can be both a subset and a proper subset. Also, every proper subset unconditionally happens to be a subset.
Answer:
Answer:
If A = {1, 3, 5, 7, 9} and B = {2, 4, 6, 8, 10}
Then U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} will be taken as a universal set.