How To Find Subsets Of A Set

Subset Calculator

Created by Anna Szczepanek , PhD

Reviewed by

Dominik Czernia , PhD candidate and Jack Bowater

Last updated:

February xv, 2022

Table of contents:

What is a subset of a gear up?
What is a proper subset?
How to use this subset calculator?
Example of how to discover subsets and proper subsets
Number of subsets and proper subsets of a set
Example of how to find the number of subsets

This subset computer can generate all the subsets of a given set, likewise as detect the total number of subsets. It tin can as well count the number of proper subsets based on the number of elements your prepare has, or maybe yous need to know how many subsets in that location are with a specific number of elements? No trouble! Our subset calculator is here to help you.

What is a subset of a set? And what is a proper subset? If yous want to larn what these terms mean, read the article beneath where nosotros requite the subset and proper subset definitions. We also explain the subset vs. proper subset distinction and show how to observe subsets and proper subsets of a set. Equally a bonus, we volition and then tell you lot what a power fix is, likewise every bit presenting to yous all the required formulas 😊

Subsets play an important function in statistics whenever you need to detect the probability of a sure event. Y'all might need information technology when working with combinations or permutations.

What is a subset of a prepare?

Subset definition:

Allow A and B be 2 sets. We say that A is a subset of B if every element of A is besides an element of B. In other words, A consists of some (possibly all) of the elements of B, only doesn't accept any elements that B doesn't take. If A is a subset of B, we tin can likewise say that B is a superset of A.

Examples:

The empty set ∅ is a subset of any set;
{1,2} is a subset of {1,two,3,four};
∅, {ane} and {1,two} are iii different subsets of {1,2}; and
Prime numbers and odd numbers are both subsets of the set of integers.

Power set definition:

The prepare of all subsets of a set (including the empty set and the set itself!) is called the ability prepare of a gear up. We commonly denote the power set of whatever set A by P(A). Note, that the ability gear up consists of sets; in particular, the elements of A are Non the elements of P(A)!

Examples:

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

As you can encounter in the examples, the ability 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 only 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} and {ii} are proper subsets of {one,2};
The empty set ∅ is a proper subset of {1,2};
Only {1,2} is NOT a proper subset of {1,two}; and
Prime numbers and odd numbers are ii distinct proper subsets of the set of all integers.

Subset vs proper subset facts:

At that place's no set without a subset. Each set has at to the lowest degree i subset: the empty set ∅;
For each set there is just one subset which is NOT a proper subset: the prepare itself;
In that location is exactly one prepare with no proper subsets: the empty set; and
Every non-empty set has at to the lowest degree ii subsets (itself and the empty set) and at to the lowest degree 1 proper subset (the empty set).

Every bit a effect, each prepare has one more subset than information technology has proper subsets. How many exactly? Check below.

Notation issue:

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

A ⊆ B we read every bit 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 nosotros read as A is a subset of B; and
C ⊊ B we read every bit C is a proper subset of B

Best stick to the convention introduced past your instructor. If yous're unsure, and want to be on the safety side, employ ⊆ for subsets and ⊊ for proper subsets: the tiny equal/unequal sign at the lesser of the symbol indicates that the subset tin/cannot exist equal to the gear up, 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 listing of subsets of a given set. Alternatively, y'all can apply it to determine the number of subsets based on the number of elements in your set up. Hither's a quick set of instruction on how to use it:

The subset calculator has 2 modes: set elements mode and gear up cardinality fashion.
For set elements style: enter the elements of your set. Initially, yous will see three fields, but more will pop up when you lot need them. You may enter up to x elements. Nosotros and so 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 gear up consists of letters, or any other elements, don't worry - supervene upon them with any numbers you want. For readability, nosotros recommend picking smaller numbers rather than larger, just, in the end, it's up to your creativity. Only call up to map the distinct elements of your set to singled-out numbers!
For ready cardinality mode: "prepare 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 x elements), the computer displays the number of subsets with all possible cardinalities; and
For larger sets (more than than ten elements), you need to enter the cardinality for which you want the subsets counted.

Tip: In both modes you lot 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}

At that place tin't be a subset with more than than four elements, as A itself has only 4 elements (a subset of A must not contain any element which is non in A). So, we listed all possible subsets of A: there are 16 of them.

Amidst them in that location is 1 subset of A which is Not a proper subset of A: A itself.
Therefore, apart from {a, b, c, d}, the subsets listed in a higher place are all possible proper subsets of A. There are 15 of them.

It'southward not difficult, is it? Merely our set had just 4 elements. What if we were to notice all the subsets of the ready {a, b, c, ..., z} containing all twenty-six letters from the English alphabet? In the side by side 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

Formula to discover the number of subsets:

If a gear up contains northward elements, then the number of subsets of this fix is equal to 2ⁿ .

To understand this formula, allow's follow this train of thought. Note, that to construct a subset for each element of the original prepare y'all take to decide whether this element will exist included in the subset or not, therefore you have ii possibilities for a given chemical element. So, in total, you have 2 * two * ... * ii possibilities, where the number of two's corresponds to the number of elements in the set, so at that place are due north of them.

Formula to find the number of proper subsets:

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

The but subset which is not proper is the set itself. Then, to get the number of proper subsets, yous only need to decrease one from the total number of subsets.

Formula to find the number of subsets with a given cardinality

Recall that "fix cardinality" is the number of elements in a set. If a set contains northward elements, then its subsets tin can have betwixt 0 and north elements. The number of subsets with k elements, where 0 ≤ thou ≤ northward, is given by the binomial coefficient:

The symbol on the left-mitt side is read "n choose yard". The exclamation marking at the right-manus side is a factorial.

This number, sometimes denoted by C(n,thousand) or nCk, is the number of k-combinations of an n-chemical element set. That is, this is the number of ways in which one thousand singled-out elements tin can be chosen from a larger set of due north distinguishable objects, where lodge doesn't matter. To acquire more, check our combinations calculator.

Example of how to find the number of subsets

Instance 1.

Presume we have a set A with 4 elements.

Kickoff, 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 - ane = 15
Side by side, we find the number of subsets of A with a given number of elements:
- Number of subsets of A with 0 elements:
  
  4! / (0! * iv!) = 1
- Number of subsets of A with ane element:
  
  four! / (1! * 3!) = 4 / 1 = four
- Number of subsets of A with ii elements:
  
  4! / (two! * 2!) = 3 * 4 / 2 = half-dozen
- Number of subsets of A with 3 elements:
  
  4! / (three! * 1!) = 4 / 1 = iv
- Number of subsets of A with 4 elements:
  
  four! / (four! * 0!) = 1

Have a expect at those numbers: 1 4 6 iv 1. Maybe you have recognized them as the 4th row of Pascal's triangle. Indeed, for a set of north elements, the n-th row of Pascal's triangle lists how many subsets with 0, 1, ..., northward elements the gear up has!

Example two.

Now we tin 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 computer to generate the 26-th row of the Pascal'due south 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