Do Many Subsets Of X Of Size K?

Subset Figurer

Created by Anna Szczepanek , PhD

Reviewed past

Dominik Czernia , PhD candidate and Jack Bowater

Last updated:

Apr 06, 2022

Table of contents:

What is a subset of a set?
What is a proper subset?
How to utilize this subset figurer?
Example of how to observe subsets and proper subsets
Number of subsets and proper subsets of a gear up
Example of how to notice the number of subsets

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

What is a subset of a gear up? And what is a proper subset? If y'all want to learn what these terms mean, read the commodity below where we give the subset and proper subset definitions. Nosotros besides explain the subset vs. proper subset distinction and prove how to notice subsets and proper subsets of a set. Every bit a bonus, nosotros will and so tell yous what a power set is, as well as presenting to you lot all the required formulas 😊

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

What is a subset of a gear up?

Subset definition:

Let A and B be two sets. Nosotros say that A is a subset of B if every chemical 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 whatever elements that B doesn't have. If A is a subset of B, nosotros can besides say that B is a superset of A.

Examples:

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

Power fix definition:

The ready of all subsets of a set (including the empty set and the set itself!) is called the power fix of a set. We usually denote the power ready of any prepare A by P(A). Notation, that the power ready consists of sets; in item, the elements of A are Not the elements of P(A)!

Examples:

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

Equally you tin can run into in the examples, the power set always has more elements than the original set. How many? Cheque the section beneath.

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

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

Examples:

{1} and {ii} are proper subsets of {1,2};
The empty fix ∅ is a proper subset of {1,2};
But {1,2} is Not a proper subset of {1,2}; and
Prime numbers and odd numbers are two singled-out proper subsets of the ready 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 1 subset: the empty ready ∅;
For each prepare there is simply 1 subset which is NOT a proper subset: the set itself;
In that location is exactly one set with no proper subsets: the empty set; and
Every non-empty set has at least 2 subsets (itself and the empty set up) and at least i proper subset (the empty set).

As a result, each set has one more than subset than information technology has proper subsets. How many exactly? Bank check below.

Annotation outcome:

Some people use the symbol ⊆ to bespeak 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, notwithstanding, use ⊂ for subsets and ⊊ for proper subsets:

A ⊂ B nosotros 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 prophylactic side, use ⊆ for subsets and ⊊ for proper subsets: the tiny equal/unequal sign at the bottom of the symbol indicates that the subset tin/cannot be equal to the set, which leaves no infinite for any ambivalence.

How to utilise this subset calculator?

Our subset reckoner is here for you whenever you wonder how to find subsets and need to generate the list of subsets of a given set up. Alternatively, you tin use it to determine the number of subsets based on the number of elements in your set. Here's a quick fix of educational activity on how to utilise it:

The subset figurer has two modes: set elements style and prepare cardinality mode.
For set elements fashion: enter the elements of your fix. Initially, you will see iii fields, only more will pop upwardly when you need them. You may enter up to x elements. We 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 but enter numbers as elements. If your gear up consists of letters, or whatever other elements, don't worry - supplant them with any numbers you desire. For readability, we recommend picking smaller numbers rather than larger, only, in the end, it's up to your inventiveness. But recollect to map the distinct elements of your ready to distinct numbers!
For ready cardinality mode: "gear up cardinality" is the number of elements in a set. Once you lot tell us how many elements your set has, nosotros count the number of (proper) subsets and:

For smaller sets (up to 10 elements), the reckoner displays the number of subsets with all possible cardinalities; and
For larger sets (more than than x elements), you lot need to enter the cardinality for which yous want the subsets counted.

Tip: In both modes y'all can restrict the output to the subsets with a given cardinality. Likewise, make sure to check out the union and intersection calculator for farther report of set up operations.

Example of how to find subsets and proper subsets

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

The subset of A containing no elements:

∅
The subsets of A containing ane chemical 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 iii elements:

{a, b, c}; {a, b, d}; {a, c, d}; {b, c, d}
The subset of A containing iv elements:

{a, b, c, d}

There can't be a subset with more than four elements, as A itself has but four elements (a subset of A must non incorporate whatsoever chemical element which is non in A). And so, we listed all possible subsets of A: there are xvi of them.

Amid them there is i subset of A which is Non 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 fifteen of them.

It'due south not hard, is it? But our set had just four elements. What if we were to find all the subsets of the set {a, b, c, ..., z} containing all twenty-half-dozen letters from the English alphabet? In the adjacent department we explain how to calculate how many subsets at that place are in a gear up without writing them all out!

Number of subsets and proper subsets of a set

Formula to detect the number of subsets:

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

To understand this formula, permit's follow this train of thought. Note, that to construct a subset for each element of the original set you take to determine whether this element volition exist included in the subset or non, therefore you have 2 possibilities for a given element. Then, in total, you have 2 * ii * ... * two possibilities, where the number of two'south corresponds to the number of elements in the set, and so at that place are n of them.

Formula to find the number of proper subsets:

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

The but subset which is not proper is the set itself. So, to get the number of proper subsets, yous just demand to subtract one from the total number of subsets.

Formula to find the number of subsets with a given cardinality

Call up that "set cardinality" is the number of elements in a prepare. If a fix contains n elements, then its subsets tin have betwixt 0 and n elements. The number of subsets with m elements, where 0 ≤ k ≤ n, is given by the binomial coefficient:

The symbol on the left-paw side is read "n choose 1000". The exclamation marking at the correct-hand side is a factorial.

This number, sometimes denoted by C(due north,thousand) or nCk, is the number of chiliad-combinations of an n-element prepare. That is, this is the number of ways in which k distinct elements tin be chosen from a larger fix of due north distinguishable objects, where order doesn't matter. To learn more, cheque our combinations calculator.

Example of how to discover the number of subsets

Example 1.

Assume we have a set A with four elements.

First, let's calculate the number of subsets and the number of proper subsets of A:
- Number of subsets of A: two⁴ = 16
- Number of proper subsets of A: ii⁴ - i = 15
Next, nosotros 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 ane element:
  
  4! / (1! * iii!) = 4 / 1 = 4
- Number of subsets of A with 2 elements:
  
  iv! / (2! * ii!) = 3 * 4 / 2 = 6
- Number of subsets of A with 3 elements:
  
  iv! / (3! * 1!) = 4 / 1 = 4
- Number of subsets of A with four elements:
  
  4! / (4! * 0!) = 1

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

Example 2.

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'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 1

From this we immediately run 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 set up (up to 10 terms):

Accented value equation Absolute value inequalities Calculation and subtracting polynomials … 30 more