Subsets: Definition, Types, Properties and Example Questions

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.

Definition of Subsets

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

• X is a subset of Y

or

• Y is the superset of X

Points to Remember:

• Every set is the subset of itself.
• Null set or Empty set is a subset of every set.

Symbol of Subset

Mathematically or symbolically, subsets are represented by ⊆.
Write X ⊆ Y to represent that Set X is the subset of Set Y.

Example Question

Question 1:

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

Question 2:

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}

Types of Subsets

Subsets are further classified into two parts:

• Proper Subset
• Improper Subset

Let us learn about types of subsets with example questions and find out the difference.

Proper Subsets

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;

• M ⊂ N and
• M ≠ N

Proper Subset Formula

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.

For Example: If a set A = {x, y}, then calculate its number of proper subsets and write them.

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})

Improper Subsets

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:

• M ⊆ N and
• M = N

What is a Power Set?

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).

Power Set Formula

If B has n elements, then P(B) has 2n, which means power sets will have certain elements.

For Example:

If set B has {0, 1} elements than its power set will be:

2n = 22 = 4

P(B) = {{}. {0}, {1}, {0, 1}}

What is a Universal Set?

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.

For example:

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

Difference between Proper and Improper Subsets

The following table is designed to help you understand the differences between proper and improper subsets:

Note: In the above table, mentioned D is the set with n number of elements.

Properties of Subsets

• Every set is unconditionally the subset of itself. Q ⊂ Q or W ⊂ W
• Empty, null, or void set or Φ is the subset of every set.
• Set M is a subset of set N, if and only if their union is equal to set B.,
  i.e., M ⊂ N » M ∪ N = N
• Any set with n elements has 2n subsets.
• If set W is the subset of set Q and set Q is the subset of set R, then set W would be the subset of R. Mathematically, W ⊂ Q and Q ⊂ R then W ⊂ R.
• If Q ⊂ R and R ⊂ Q, then Q = R
• If a set Q is a subset of set R, we can say that R is a superset of Q
• Set M is a subset of set N, if and only if their intersection is equal to set M, i.e., M ⊂ N » M ∩ N = M.

Subsets Practice Questions

Question: Determine whether P is the subset of O if P = {a, b, c} and O = {a, b, c, d, e, f, g}

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.

Question: If set A = {a, e, i, o, u} then find its total numbers of subsets and number of proper subsets.

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

Question: If set A = {You} and set B = {Sister, You, Mother, Father}, how is A ⊂ B? Explain?

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.

Question: Find the proper and improper subsets of B = {2, 4, 6}

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}

Frequently Asked Questions

Question: How many subsets with 2, 3 and 1 elements respectively?

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.

Question: What are the classifications of subset?

Answer:
Subsets are classified into Proper subset and Improper subset.

Question: Can a subset be a proper subset?

Answer:
Yes, a set can be both a subset and a proper subset. Also, every proper subset unconditionally happens to be a subset.

Question: Explain the concept of proper and improper subsets by an example?

Answer:

• Proper subset:
  A = {x, y, z} and B = {w, x, y, z}
• Improper subset:
  A = {7, 14, 21, 28} and B = {7, 14, 21, 28}

Question: Write the universal set of the set A = {1, 3, 5, 7, 9} and set B = {2, 4, 6, 8, 10}

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.