How To Find Subsets Of A Set

Subset Estimator

Created by Anna Szczepanek , PhD

Reviewed by

Dominik Czernia , PhD candidate and Jack Bowater

Last updated:

Apr 06, 2022

Table of contents:

• What is a subset of a gear up?
• What is a proper subset?
• How to utilise this subset reckoner?
• Example of how to find subsets and proper subsets
• Number of subsets and proper subsets of a set
• Case of how to detect the number of subsets

This subset estimator can generate all the subsets of a given fix, likewise as detect the total number of subsets. It tin can besides count the number of proper subsets based on the number of elements your fix has, or maybe you lot need to know how many subsets in that location are with a specific number of elements? No problem! Our subset figurer is here to help you.

What is a subset of a set? And what is a proper subset? If you desire to learn what these terms mean, read the article below where we requite the subset and proper subset definitions. Nosotros also explicate the subset vs. proper subset stardom and show how to find subsets and proper subsets of a set. As a bonus, nosotros will then tell you what a power fix is, too 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 fix?

Subset definition:

Let A and B be 2 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, merely doesn't have whatever 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 up ∅ is a subset of any set;
2. {1,2} is a subset of {1,2,3,four};
3. ∅, {1} and {1,2} are 3 different subsets of {1,ii}; and
4. Prime number numbers and odd numbers are both subsets of the set up of integers.

Ability set definition:

The set of all subsets of a set (including the empty set and the ready itself!) is called the power set of a prepare. We normally denote the power set of whatever 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}, so P(A) = {∅, {1}, {2}, {one,two}}; and
2. P(∅) = {∅}.

As you can see in the examples, the ability set always has more elements than the original set. How many? Bank 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 simply not all of the elements of B and A doesn't have any elements that don't belong to B.

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

Examples:

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

2. The empty gear up ∅ is a proper subset of {1,ii};

3. Simply {1,two} is NOT a proper subset of {1,2}; and

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

Subset vs proper subset facts:

• At that place's no set without a subset. Each set has at least ane subset: the empty set ∅;

• For each set up there is only 1 subset which is Non a proper subset: the set itself;

• At that place 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 ready) and at least i proper subset (the empty fix).

Equally a consequence, each fix has one more subset than information technology has proper subsets. How many exactly? Cheque below.

Notation effect:

Some people utilise 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 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 instructor. If you're unsure, and want to be on the safe side, utilise ⊆ 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 ambivalence.

How to utilise this subset calculator?

Our subset calculator is here for you lot whenever yous wonder how to find subsets and need to generate the list of subsets of a given set. Alternatively, you can use information technology 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 apply it:

1. The subset calculator has two modes: set elements style and set cardinality way.

2. For set elements mode: enter the elements of your set. Initially, you lot will see three fields, just more volition popular up when you need them. You may enter upwards to 10 elements. We so count the subsets and proper subsets of your prepare. Y'all 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 messages, 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 cease, it's up to your creativity. Simply remember to map the singled-out elements of your set to distinct numbers!

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

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

• For larger sets (more than 10 elements), y'all 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. Too, make certain to check out the wedlock and intersection estimator for further written report of fix operations.

Case 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 ane 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 4 elements:

  {a, b, c, d}

There can't exist 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, nosotros listed all possible subsets of A: there are sixteen of them.

Among them there 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 above are all possible proper subsets of A. There are 15 of them.

It's not hard, is it? Only our fix had just 4 elements. What if nosotros were to notice all the subsets of the ready {a, b, c, ..., z} containing all twenty-six letters from the English alphabet? In the next department 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 up

1. Formula to find the number of subsets:

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

To understand this formula, allow's follow this railroad train of thought. Note, that to construct a subset for each element of the original set you take to make up one's mind whether this element volition exist included in the subset or not, therefore you have ii possibilities for a given chemical element. So, in full, you have 2 * 2 * ... * 2 possibilities, where the number of two'due south corresponds to the number of elements in the set, then there are northward of them.

2. Formula to discover 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. Then, to get the number of proper subsets, y'all only need to subtract one from the full number of subsets.

3. 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 prepare contains due north elements, then its subsets tin have betwixt 0 and n elements. The number of subsets with k elements, where 0 ≤ yard ≤ n, is given by the binomial coefficient:

The symbol on the left-hand side is read "n cull thou". The exclamation marker at the right-mitt side is a factorial.

This number, sometimes denoted by C(n,k) or nCk, is the number of grand-combinations of an northward-chemical element ready. That is, this is the number of ways in which m distinct elements can be called from a larger gear up of n distinguishable objects, where order doesn't affair. To learn more, check our combinations reckoner.

Example of how to find the number of subsets

Example ane.

Assume we have a fix 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: ii⁴ - i = 15

2. Adjacent, we observe the number of subsets of A with a given number of elements:

   • Number of subsets of A with 0 elements:

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

   • Number of subsets of A with 1 element:

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

   • Number of subsets of A with 2 elements:

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

   • Number of subsets of A with 3 elements:

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

   • Number of subsets of A with 4 elements:

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

Accept a look at those numbers: i 4 6 4 ane. Maybe you have recognized them as the fourth row of Pascal's triangle. Indeed, for a set of n elements, the north-th row of Pascal's triangle lists how many subsets with 0, one, ..., north elements the set up 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.
Every bit it has 26 elements, we use the Pascal's triangle reckoner 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 run across that {a, b, ..., z} has

• 1 subset with 0 elements

• 26 subsets with 1 element

• 325 subsets with two elements

• 2600 subsets with iii elements

  ...

• 10400600 subsets with 13 elements!

  ...

In total, there are 67108864 subsets!

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

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

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

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