How To Find The Number Of Subsets In A Set

Subset Estimator

Created by Anna Szczepanek , PhD

Reviewed by

Dominik Czernia , PhD candidate and Jack Bowater

Last updated:

Apr 06, 2022

Tabular array 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 reckoner can generate all the subsets of a given set, likewise as observe the total number of subsets. It tin also count the number of proper subsets based on the number of elements your set up has, or maybe you need to know how many subsets in that location are with a specific number of elements? No problem! Our subset calculator is here to assist you.

What is a subset of a set? And what is a proper subset? If you want to larn what these terms hateful, read the commodity below where we give the subset and proper subset definitions. Nosotros also explain the subset vs. proper subset distinction and show how to find subsets and proper subsets of a gear up. Every bit a bonus, we will then tell yous what a power gear up is, as well as presenting to you all the required formulas 😊

Subsets play an of import part in statistics whenever you demand to observe the probability of a certain effect. You might demand it when working with combinations or permutations.

What is a subset of a set?

Subset definition:

Let A and B exist two 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 (maybe all) of the elements of B, but doesn't have whatsoever elements that B doesn't accept. If A is a subset of B, nosotros can likewise say that B is a superset of A.

Examples:

1. The empty set ∅ is a subset of any set;
2. {i,ii} is a subset of {1,2,3,4};
3. ∅, {one} and {1,2} are three different subsets of {ane,two}; and
4. Prime numbers and odd numbers are both subsets of the gear up of integers.

Power set definition:

The ready of all subsets of a set (including the empty set and the set itself!) is called the power set of a ready. We unremarkably denote the power ready of any set A past 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 = {i,2}, so P(A) = {∅, {1}, {2}, {1,ii}}; and
2. P(∅) = {∅}.

Every bit you tin come across in the examples, the power set always has more than elements than the original set. How many? Cheque 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 non all of the elements of B and A doesn't have any elements that don't vest to B.

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

Examples:

1. {one} and {2} are proper subsets of {ane,two};

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

3. But {one,2} is Non a proper subset of {1,2}; and

4. Prime numbers and odd numbers are two singled-out proper subsets of the prepare of all integers.

Subset vs proper subset facts:

• There's no set up without a subset. Each ready has at least ane subset: the empty ready ∅;

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

• At that place is exactly ane set with no proper subsets: the empty prepare; and

• Every not-empty ready has at least two subsets (itself and the empty set) and at least 1 proper subset (the empty set).

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

Notation result:

Some people utilize the symbol ⊆ to signal a subset and ⊂ to signal a proper subset:

• A ⊆ B nosotros read equally A is a subset of B; and
• C ⊂ B nosotros read every bit 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

All-time 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 fix, which leaves no space for any ambiguity.

How to utilize this subset calculator?

Our subset calculator is here for you whenever yous wonder how to find subsets and need to generate the listing of subsets of a given ready. Alternatively, you tin can use information technology to determine the number of subsets based on the number of elements in your prepare. Here'due south a quick gear up of didactics on how to use it:

1. The subset computer has two modes: gear up elements way and gear up cardinality mode.

2. For gear up elements mode: enter the elements of your set. Initially, you lot will come across three fields, but more than will popular up when you need them. You may enter upwards to 10 elements. We then count the subsets and proper subsets of your set. You can also display the listing of subsets with the number of elements of your choosing.

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

3. For set cardinality manner: "prepare cardinality" is the number of elements in a set up. One time 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), yous need to enter the cardinality for which you lot want the subsets counted.

Tip: In both modes you tin can restrict the output to the subsets with a given cardinality. Also, brand sure to check out the matrimony and intersection figurer for farther 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 3 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 4 elements, as A itself has only iv elements (a subset of A must not comprise any element which is not in A). So, we listed all possible subsets of A: there are sixteen of them.

Amid them there is one subset of A which is Non a proper subset of A: A itself.
Therefore, apart from {a, b, c, d}, the subsets listed to a higher place are all possible proper subsets of A. There are fifteen of them.

Information technology's not hard, is it? But our gear up had simply 4 elements. What if we were to find all the subsets of the set {a, b, c, ..., z} containing all 20-six letters from the English alphabet? In the adjacent department nosotros explain how to summate how many subsets at that place 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 prepare contains n elements, then the number of subsets of this set is equal to 2ⁿ .

To empathise this formula, let'due south follow this railroad train of thought. Annotation, that to construct a subset for each element of the original set you take to make up one's mind whether this element will be included in the subset or not, therefore yous accept two possibilities for a given chemical element. So, in total, you lot have ii * 2 * ... * 2 possibilities, where the number of two's corresponds to the number of elements in the set, then in that location are n of them.

2. Formula to notice the number of proper subsets:

If a set contains northward elements, so the number of subsets of this fix is equal to 2ⁿ - i.

The only subset which is non proper is the fix itself. And then, to get the number of proper subsets, you but need to subtract one from the total number of subsets.

3. Formula to detect the number of subsets with a given cardinality

Retrieve that "set cardinality" is the number of elements in a set up. If a prepare contains n elements, then its subsets can have between 0 and n elements. The number of subsets with chiliad elements, where 0 ≤ k ≤ n, is given by the binomial coefficient:

The symbol on the left-paw side is read "n choose k". The exclamation mark at the correct-hand side is a factorial.

This number, sometimes denoted past C(northward,1000) or nCk, is the number of k-combinations of an n-chemical element set. That is, this is the number of ways in which k singled-out elements can be chosen from a larger ready of northward distinguishable objects, where order doesn't matter. To learn more, check our combinations reckoner.

Example of how to detect the number of subsets

Case 1.

Assume nosotros 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: two⁴ - ane = 15

2. 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:

     four! / (0! * 4!) = one

   • Number of subsets of A with 1 element:

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

   • Number of subsets of A with 2 elements:

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

   • Number of subsets of A with 3 elements:

     iv! / (3! * 1!) = iv / 1 = four

   • Number of subsets of A with iv elements:

     4! / (4! * 0!) = i

Accept a look at those numbers: i 4 half dozen 4 ane. Maybe you have recognized them as the quaternary row of Pascal's triangle. Indeed, for a ready of north elements, the n-th row of Pascal'southward triangle lists how many subsets with 0, 1, ..., n elements the set has!

Instance 2.

Now we can finally go back to the set up {a, b, c, ..., z} of all the letters of the English alphabet.
As information technology has 26 elements, nosotros use the Pascal'south triangle calculator to generate the 26-th row of the Pascal'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 i

From this nosotros immediately encounter that {a, b, ..., z} has

• ane subset with 0 elements

• 26 subsets with ane element

• 325 subsets with ii elements

• 2600 subsets with 3 elements

  ...

• 10400600 subsets with 13 elements!

  ...

In total, in that location are 67108864 subsets!

Enter the elements of your set (up to 10 terms):

Absolute value equation Absolute value inequalities Adding and subtracting polynomials … 32 more

Source: https://www.omnicalculator.com/math/subset

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