How To Find The Number Of Subsets

Subset Calculator

Created by Anna Szczepanek , PhD

Reviewed by

Dominik Czernia , PhD and Jack Bowater

Final updated:

December 02, 2022

This subset calculator tin can generate all the subsets of a given set, also every bit observe the total number of subsets. It tin can also count the number of proper subsets based on the number of elements your fix has, or perhaps you need to know how many subsets there are with a specific number of elements? No problem! Our subset calculator is hither to help yous.

What is a subset of a set? And what is a proper subset? If you want to acquire what these terms mean, read the article below, where we give the subset and proper subset definitions. We too explicate the subset vs. proper subset distinction and testify how to find subsets and proper subsets of a gear up. As a bonus, we will and so tell you what a power prepare is, equally well every bit present to you all the required formulas 😊

Subsets play an important office in statistics whenever you need to find the probability of a sure effect. Yous might need information technology when working with combinations or permutations.

What is a subset of a set?

Subset definition:

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

Examples:

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

Power set definition:

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

Examples:

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

Equally y'all can run into in the examples, the power set always has more elements than the original set. How many? Check the department below. And if you lot'd similar to learn even more than most this type of set up, the ability set up calculator may satisfy your curiosity!

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 merely not all of the elements of B and A doesn't take any elements that don't belong to B.

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

Examples:

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

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

3. But {1,ii} is Not a proper subset of {1,2}; and

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

Subset vs proper subset facts:

• There'south no set without a subset. Each set has at to the lowest degree one subset: the empty ready ∅;

• For each set there is but 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 not-empty set has at to the lowest degree ii subsets (itself and the empty set) and at least i proper subset (the empty set).

As a consequence, each set has ane more than subset than it has proper subsets. How many exactly? Bank check below.

Notation issue:

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

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

Others, however, apply ⊂ 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 yous're unsure, and want to be on the safe side, employ ⊆ for subsets and ⊊ for proper subsets: the tiny equal/unequal sign at the lesser of the symbol indicates that the subset can/cannot be equal to the prepare, which leaves no space for any ambivalence.

How to use this subset calculator?

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

1. The subset figurer has two modes: ready elements mode and set cardinality mode.

2. For set elements mode: enter the elements of your set. Initially, you will see three fields, just more will popular upwardly when yous demand them. You may enter up to 10 elements. We so count the subsets and proper subsets of your set up. You tin can likewise display the list of subsets with the number of elements of your choosing.

   Yous tin can just enter numbers every bit elements. If your fix consists of messages, or any other elements, don't worry - replace them with any numbers y'all want. For readability, we recommend picking smaller numbers rather than larger, simply, in the end, it's up to your creativity. Just think to map the distinct elements of your set to distinct numbers!

3. For set cardinality mode: "gear up cardinality" is the number of elements in a set. Once you tell u.s. how many elements your fix has, nosotros 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 10 elements), you need to enter the cardinality for which you want the subsets counted.

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

Instance of how to find subsets and proper subsets

Let united states list all subsets of A = {a, b, c, d}.

• The subset of A containing no elements:

  ∅

• The subsets of A containing i 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 iv elements:

  {a, b, c, d}

In that location can't be a subset with more four elements, as A itself has but 4 elements (a subset of A must not contain any chemical element which is non in A). And then, nosotros listed all possible subsets of A: at that place are 16 of them.

Amongst them there is one subset of A which is NOT a proper subset of A: A itself.
Therefore, autonomously from {a, b, c, d}, the subsets listed to a higher place are all possible proper subsets of A. In that location are 15 of them.

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

Number of subsets and proper subsets of a set up

1. Formula to find the number of subsets:

If a set up contains n elements, and so the number of subsets of this set is equal to 2ⁿ .

To understand this formula, permit'south follow this train of thought. Note, that to construct a subset for each element of the original ready yous take to decide whether this element will be included in the subset or not, therefore you lot accept 2 possibilities for a given element. So, in total, yous take 2 * ii * ... * two possibilities, where the number of 2's corresponds to the number of elements in the set, and so there are due north of them.

2. Formula to notice the number of proper subsets:

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

The only subset which is not proper is the set up itself. Then, to get the number of proper subsets, you lot but need to subtract 1 from the total number of subsets.

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

Call up that "fix cardinality" is the number of elements in a gear up. If a prepare contains north elements, and so its subsets can accept between 0 and due north elements. The number of subsets with one thousand elements, where 0 ≤ g ≤ north, is given past the binomial coefficient:

The symbol on the left-mitt side is read "north choose k". The exclamation marker at the right-paw side is a factorial.

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

Instance of how to find the number of subsets

Example i.

Presume we have a set up A with 4 elements.

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

   • Number of subsets of A: 2⁴ = sixteen

   • 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:

     four! / (0! * iv!) = i

   • Number of subsets of A with 1 element:

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

   • Number of subsets of A with 2 elements:

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

   • Number of subsets of A with 3 elements:

     4! / (iii! * 1!) = 4 / 1 = four

   • Number of subsets of A with 4 elements:

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

Have a await at those numbers: 1 4 6 4 1. Maybe you lot have recognized them every bit the quaternary row of Pascal's triangle. Indeed, for a ready of northward elements, the due north-th row of Pascal's triangle lists how many subsets with 0, ane, ..., n elements the fix has!

Example 2.

Now we tin can finally get dorsum to the ready {a, b, c, ..., z} of all the letters of the English language alphabet.
As it has 26 elements, we use the Pascal'southward triangle computer to generate the 26-th row of the Pascal'south triangle:

i 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 come across 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!

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

Absolute value equation Accented value inequalities Calculation and subtracting polynomials … 34 more

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

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