# Subset Calculator

By Anna Szczepanek, PhD
Last updated: Jun 28, 2021

This subset calculator can generate all the subsets of a given set, as well as find the total number of subsets. It can also count the number of proper subsets based on the number of elements your set has, or maybe you need to know how many subsets there are with a specific number of elements? No problem! Our subset calculator is here to help you.

What is a subset of a set? And what is a proper subset? If you want to learn what these terms mean, read the article below where we give the subset and proper subset definitions. We also explain the subset vs. proper subset distinction and show how to find subsets and proper subsets of a set. As a bonus, we will then tell you what a power set is, as well as presenting to you all the required formulas 😊

Subsets play an important role in statistics whenever you need to find the probability of a certain event. You might need it when working with combinations or permutations.

## What is a subset of a set?

#### Subset definition:

Let A and B be two sets. We say that A is a subset of B if every element of A is also an element of B. In other words, A consists of some (possibly all) of the elements of B, but doesn't have any elements that B doesn't have. If A is a subset of B, we can also say that B is a superset of A.

Examples:

1. The empty set `∅` is a subset of any set;
2. `{1,2}` is a subset of `{1,2,3,4}`;
3. `∅`, `{1}` and `{1,2}` are three different subsets of `{1,2}`; and
4. Prime numbers and odd numbers are both subsets of the set of integers.

#### Power set definition:

The set of all subsets of a set (including the empty set and the set itself!) is called the power set of a set. We usually denote the power set of any set A by P(A). Note, that the power set consists of sets; in particular, the elements of A are NOT the elements of P(A)!

Examples:

1. If `A = {1,2}`, then `P(A) = {∅, {1}, {2}, {1,2}}`; and
2. `P(∅) = {∅}`.

As you can see in the examples, the power set always has more elements than the original set. How many? Check the section below.

## What is a proper subset?

#### Proper subset definition:

A is a proper subset of B if A is a subset of B and A isn't equal to B. In other words, A has some but not all of the elements of B and A doesn't have any elements that don't belong to B.

We can also say that B is a proper superset of A.

Examples:

1. `{1}` and `{2}` are proper subsets of `{1,2}`;

2. The empty set `∅` is a proper subset of `{1,2}`;

3. But `{1,2}` is NOT a proper subset of `{1,2}`; and

4. Prime numbers and odd numbers are two distinct proper subsets of the set of all integers.

#### Subset vs proper subset facts:

• There's no set without a subset. Each set has at least one subset: the empty set `∅`;

• For each set there is only one subset which is NOT a proper subset: the set itself;

• There is exactly one set with no proper subsets: the empty set; and

• Every non-empty set has at least two subsets (itself and the empty set) and at least one proper subset (the empty set).

As a consequence, each set has one more subset than it has proper subsets. How many exactly? Check below.

#### Notation issue:

Some people use the symbol ⊆ to indicate a subset and ⊂ to indicate a proper subset:

• `A ⊆ B` we read as A is a subset of B; and
• `C ⊂ B` we read as C is a proper subset of B

Others, however, use ⊂ for subsets and ⊊ for proper subsets:

• `A ⊂ B` we read as A is a subset of B; and
• `C ⊊ B` we read as C is a proper subset of B

Best stick to the convention introduced by your teacher. If you're unsure, and want to be on the safe side, use ⊆ for subsets and ⊊ for proper subsets: the tiny equal/unequal sign at the bottom of the symbol indicates that the subset can/cannot be equal to the set, which leaves no space for any ambiguity.

## How to use this subset calculator?

Our subset calculator is here for you whenever you wonder how to find subsets and need to generate the list of subsets of a given set. Alternatively, you can use it to determine the number of subsets based on the number of elements in your set. Here's a quick set of instruction on how to use it:

1. The subset calculator has two modes: `set elements` mode and `set cardinality` mode.

2. For `set elements` mode: enter the elements of your set. Initially, you will see three fields, but more will pop up when you need them. You may enter up to 10 elements. We then count the subsets and proper subsets of your set. You can also display the list of subsets with the number of elements of your choosing.

You can only enter numbers as elements. If your set consists of letters, or any other elements, don't worry - replace them with any numbers you want. For readability, we recommend picking smaller numbers rather than larger, but, in the end, it's up to your creativity. Just remember to map the distinct elements of your set to distinct numbers!

3. For `set cardinality` mode: "set cardinality" is the number of elements in a set. Once you tell us how many elements your set has, we count the number of (proper) subsets and:

• For smaller sets (up to ten elements), the calculator displays the number of subsets with all possible cardinalities; and

• For larger sets (more than ten elements), you need to enter the cardinality for which you want the subsets counted.

Tip: In both modes you can restrict the output to the subsets with a given cardinality. Also, make sure to check out the union and intersection calculator for further study of set operations.

## Example of how to find subsets and proper subsets

Let us list all subsets of `A = {a, b, c, d}`.

• The subset of `A` containing no elements:

`∅`

• The subsets of `A` containing one element:

`{a}; {b}; {c}; {d}`

• The subsets of `A` containing two elements:

`{a, b}; {a, c}; {a, d}; {b, c}; {b, d}; {c, d}`

• The subsets of `A` containing three elements:

`{a, b, c}; {a, b, d}; {a, c, d}; {b, c, d}`

• The subset of `A` containing four elements:

`{a, b, c, d}`

There can't be a subset with more than four elements, as `A` itself has only four elements (a subset of `A` must not contain any element which is not in `A`). So, we listed all possible subsets of `A`: there are `16` of them.

Among them there is one subset of `A` which is NOT a proper subset of `A`: `A` itself. Therefore, apart from `{a, b, c, d}`, the subsets listed above are all possible proper subsets of `A`. There are `15` of them.

It's not hard, is it? But our set had just `4` elements. What if we were to find all the subsets of the set `{a, b, c, ..., z}` containing all twenty-six letters from the English alphabet? In the next section we explain how to calculate how many subsets there are in a set without writing them all out!

## Number of subsets and proper subsets of a set

1. Formula to find the number of subsets:

If a set contains `n` elements, then the number of subsets of this set is equal to `2ⁿ`.

To understand this formula, let's follow this train of thought. Note, that to construct a subset for each element of the original set you have to decide whether this element will be included in the subset or not, therefore you have two possibilities for a given element. So, in total, you have `2 * 2 * ... * 2` possibilities, where the number of two's corresponds to the number of elements in the set, so there are `n` of them.

1. Formula to find the number of proper subsets:

If a set contains `n` elements, then the number of subsets of this set is equal to `2ⁿ - 1`.

The only subset which is not proper is the set itself. So, to get the number of proper subsets, you just need to subtract one from the total number of subsets.

1. Formula to find the number of subsets with a given cardinality

Recall that "set cardinality" is the number of elements in a set. If a set contains `n` elements, then its subsets can have between `0` and `n` elements. The number of subsets with `k` elements, where `0 ≤ k ≤ n`, is given by the binomial coefficient: The symbol on the left-hand side is read "n choose k". The exclamation mark at the right-hand side is a factorial.

This number, sometimes denoted by `C(n,k)` or `nCk`, is the number of `k`-combinations of an `n`-element set. That is, this is the number of ways in which `k` distinct elements can be chosen from a larger set of `n` distinguishable objects, where order doesn't matter. To learn more, check our combinations calculator.

## Example of how to find the number of subsets

Example 1.

Assume we have a set `A` with `4` elements.

1. First, let's calculate the number of subsets and the number of proper subsets of `A`:

• Number of subsets of `A`: `2⁴ = 16`

• Number of proper subsets of `A`: `2⁴ - 1 = 15`

2. Next, we find the number of subsets of `A` with a given number of elements:

• Number of subsets of `A` with `0` elements:

`4! / (0! * 4!) = 1`

• Number of subsets of `A` with `1` element:

`4! / (1! * 3!) = 4 / 1 = 4`

• Number of subsets of `A` with `2` elements:

`4! / (2! * 2!) = 3 * 4 / 2 = 6`

• Number of subsets of `A` with `3` elements:

`4! / (3! * 1!) = 4 / 1 = 4`

• Number of subsets of `A` with `4` elements:

`4! / (4! * 0!) = 1`

Take a look at those numbers: `1 4 6 4 1`. Maybe you have recognized them as the fourth row of Pascal's triangle. Indeed, for a set of `n` elements, the `n`-th row of Pascal's triangle lists how many subsets with `0, 1, ..., n` elements the set has!

Example 2.

Now we can finally get back to the set `{a, b, c, ..., z}` of all the letters of the English alphabet. As it has `26` elements, we use the Pascal's triangle calculator to generate the 26-th row of the Pascal's triangle:

`1 26 325 2600 14950 65780 230230 657800 1562275 3124550 5311735 7726160 9657700 10400600 9657700 7726160 5311735 3124550 1562275 657800 230230 65780 14950 2600 325 26 1`

From this we immediately see that `{a, b, ..., z}` has

• `1` subset with `0` elements

• `26` subsets with `1` element

• `325` subsets with `2` elements

• `2600` subsets with `3` elements

...

• `10400600` subsets with `13` elements!

...

In total, there are `67108864` subsets!

Anna Szczepanek, PhD
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