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Subset Calculator

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Our subset calculator is a handy tool that assists users in calculating and generating all possible subsets or smaller groups that can be formed from larger groups or sets of items. It makes it easier to organize and analyze data or solve mathematical problems involving subsets, all while working within the limits of the original group of items.

What is a Subset of a Set?

In a Set,

“A subset is a group of elements that are drawn from the Set, and each element of the Subset is also part of the original Set”.

If A and B are two sets, and all the elements of set B also exist in set A then set B is called a subset of set A, and set A is called a superset of B.

Subset Symbols:

There are two types of subset symbols such as:

  • ⊆ =  "is a subset of" or (equal to)
  • ⊂ =  “is a proper subset of” (means a subset of but NOT equal to)

In Mathematics, the subsets are expressed by the symbol “⊆” or “⊂” and pronounced by a  “Subset Notation”.

Types of Subsets:

There are usually two types of subsets exist such as:

1. Proper Subset:

A proper subset is when all the elements in one set (let's call it Set A) also exist in another set (Set B), but Set A must be missing at least one item not in Set B. In other words, Set A is a tinny part of Set B.

For Example:

  • Set A: {apples, bananas, oranges}
  • Set B: {apples, bananas, oranges, grapes}

In this case, Set A is a proper subset of Set B because all the items in Set A are also in Set B, but Set B has one extra item (grapes) that Set A doesn't have.

Proper Subset Symbols:

The symbolic representation of a proper subset is “⊂”. eg., in above discussed example if Set A is a proper subset of Set B, then we will write as:

  • A ⊂ B or
  • A ≠ B

Proper Subsets Formula:

If a set contains “n” elements, then the number of the subset for a particular set is:

$$ 2^n $$

The No. of proper subsets of the given subset can be calculated by the formula 

$$ 2^n-1 $$

2. Improper Subset:

The term improper subset refers to a subset that contains all the elements of the original set.

Example 1:

If Set A= {1,2,3} Then, the subsets of A are;

{}, {1}, {2}, {3}, {1,2}, {2,3}, {1,3} and {1,2,3}.

Where, {}, {1}, {2}, {3}, {1,2}, {2,3}, {1,3} are the proper subsets & {1,2,3} is the improper subsets. Therefore, we can write {1,2,3} ⊆ A. You can also find these values from the subset calculator.

Example 2:

An improper subset is like having a pizza slice (Set P), exactly the same as another slice from a different pizza (Set Q), with all the same ingredients and toppings, then P is the improper subset of Q here. In simple words, there's no difference between the P and Q.

Improper Subset Symbols:

The symbol used to write the improper subsets is “⊆”. eg., in above discussed example if P is an improper subset of Q, then we will write as: 

  • P ⊆ Q or 
  • P = Q 

Improper Subset Formula:

If you have 'n' elements in a set, the number of improper subsets is always '1'. In simple words, the number of subsets of a set does not depend on the number of elements.

How to Calculate the Subsets?

You can calculate the number of subsets manually by $$ 2^n $$


Let’s say for a set X with the elements, X = {2, 4}, calculate the Proper Subset.

The formula for the proper subset: $$ 2^n-1 $$


n = The total number of elements in a set.

X = {2, 4}

The total number of elements (n) in the set= 2

Therefore,  the number of proper subsets:

$$ X = 2^n-1 $$

$$ X = 2^2-1 $$

$$ X = 4 - 1 $$

$$ X = 3 $$

Therefore the total number of proper subsets for the given set is { }, {1}, and {2} and can be denoted as { }, {1}, {2} ⊂ X. All these values could also be calculated simultaneously by subsets calculator.


In case you want to calculate the Improper Subset of the above-discussed example, in that scenario, the improper subset is: 

{2, 4} and can be denoted as {2, 4} ⊆ X.

Steps to Use Subset Calculator:

Using this free subset calculator is very easy and convenient, you just need to put some inputs for this such as:

What to do:

  • Select the option you want to calculate
  • Enter all the set values separated with a comma.
  • Press the Calculate button

What you get:

  • The values of subsets and proper subsets.
  • The number of subsets


What is meant by a Power Set?

All the subsets including the original set itself & empty set are known as power sets and it is normally expressed as P.

What is the difference between a subset vs proper subset?

The following table summarizes the differences between proper subset vs subset:

Subset Proper Subset
If A is a subset of B, we can write it as A ⊆ B. If A is a proper subset of B, we can write it as A ⊊ B (or) A ⊂ B.
If A ⊆ B, then A is a subset of B and A may or may not be equal to B. If A ⊂ B, then A is a subset of B but A is NOT equal to B.
(i) {1, 2, 3} is a subset of {1, 2, 3}.

(ii) {1, 2, 3} is a subset of {1, 2, 3, 4}
(i) {1, 2, 3} is NOT a proper subset of {1, 2, 3}.

(ii) {1, 2, 3} is a proper subset of {1, 2, 3, 4}

References: Subset, Proper Subset, Proper Subset Symbol, Proper Subset Formula, Improper Subset, and Improper Subset Formula

Alan Walker

Studies mathematics sciences, and Technology. Tech geek and a content writer. Wikipedia addict who wants to know everything. Loves traveling, nature, reading. Math and Technology have done their part, and now it's the time for us to get benefits.

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