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Wolfram Language & System Documentation Center
Subsets
  • See Also
    • Tuples
    • IntegerDigits
    • Binomial
    • IntegerPartitions
    • SymmetricPolynomial
    • BellB
    • RandomSample
    • SubsetQ
    • Subsequences
    • Groupings
    • Permutations
  • Related Guides
    • Discrete Mathematics
    • Operations on Sets
    • Finite Mathematics
    • Constructing Lists
    • Incrementals
    • Rearranging & Restructuring Lists
    • Math & Counting Operations on Lists
  • Tech Notes
    • Lists as Sets
    • See Also
      • Tuples
      • IntegerDigits
      • Binomial
      • IntegerPartitions
      • SymmetricPolynomial
      • BellB
      • RandomSample
      • SubsetQ
      • Subsequences
      • Groupings
      • Permutations
    • Related Guides
      • Discrete Mathematics
      • Operations on Sets
      • Finite Mathematics
      • Constructing Lists
      • Incrementals
      • Rearranging & Restructuring Lists
      • Math & Counting Operations on Lists
    • Tech Notes
      • Lists as Sets

Subsets[list]

gives a list of all possible subsets of list.

Subsets[list,n]

gives all subsets containing at most n elements.

Subsets[list,{n}]

gives all subsets containing exactly n elements.

Subsets[list,{nmin,nmax}]

gives all subsets containing between nmin and nmax elements.

Subsets[list,nspec,s]

limits the result to the first s subsets.

Subsets[list,nspec,{s}]

gives if possible the s ^(th) subset.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Generalizations & Extensions  
Applications  
Properties & Relations  
Possible Issues  
Neat Examples  
See Also
Tech Notes
Related Guides
Related Links
History
Cite this Page
BUILT-IN SYMBOL
  • See Also
    • Tuples
    • IntegerDigits
    • Binomial
    • IntegerPartitions
    • SymmetricPolynomial
    • BellB
    • RandomSample
    • SubsetQ
    • Subsequences
    • Groupings
    • Permutations
  • Related Guides
    • Discrete Mathematics
    • Operations on Sets
    • Finite Mathematics
    • Constructing Lists
    • Incrementals
    • Rearranging & Restructuring Lists
    • Math & Counting Operations on Lists
  • Tech Notes
    • Lists as Sets
    • See Also
      • Tuples
      • IntegerDigits
      • Binomial
      • IntegerPartitions
      • SymmetricPolynomial
      • BellB
      • RandomSample
      • SubsetQ
      • Subsequences
      • Groupings
      • Permutations
    • Related Guides
      • Discrete Mathematics
      • Operations on Sets
      • Finite Mathematics
      • Constructing Lists
      • Incrementals
      • Rearranging & Restructuring Lists
      • Math & Counting Operations on Lists
    • Tech Notes
      • Lists as Sets

Subsets

Subsets[list]

gives a list of all possible subsets of list.

Subsets[list,n]

gives all subsets containing at most n elements.

Subsets[list,{n}]

gives all subsets containing exactly n elements.

Subsets[list,{nmin,nmax}]

gives all subsets containing between nmin and nmax elements.

Subsets[list,nspec,s]

limits the result to the first s subsets.

Subsets[list,nspec,{s}]

gives if possible the s ^(th) subset.

Details

  • Subsets[list] gives the power set of list.
  • Subsets[list] orders subsets with shortest first, and later elements in list omitted first.
  • If the elements of list are in the order returned by Sort, then the complete result from Subsets[list] will also be in this order.
  • Subsets[list,All] is equivalent to Subsets[list].
  • Subsets[list,{nmin,nmax,dn}] gives subsets containing nmin, nmin+dn, … elements.
  • Subsets[list,nspec,spec] gives the same result as Take[Subsets[list,nspec],spec], provided that the elements specified by spec are present.

Examples

open all close all

Basic Examples  (2)

All possible subsets (power set):

Wolfram Language code: Subsets[{a, b, c}]

All possible subsets containing up to 2 elements:

Wolfram Language code: Subsets[{a, b, c, d}, 2]

Subsets containing exactly 2 elements:

Wolfram Language code: Subsets[{a, b, c, d}, {2}]

Scope  (4)

The first 5 subsets containing 3 elements:

Wolfram Language code: Subsets[{a, b, c, d, e}, {3}, 5]

All subsets with even length:

Wolfram Language code: Subsets[{a, b, c, d, e}, {0, 5, 2}]

The 69381^(st) subset:

Wolfram Language code: Subsets[Range[20], All, {69381}]

All subsets of {a,b,c,d}:

Wolfram Language code: Subsets[{a, b, c, d}]

The odd-numbered subsets of {a,b,c,d} in reverse order:

Wolfram Language code: Subsets[{a, b, c, d}, All, {15, 1, -2}]

Generalizations & Extensions  (1)

Use any head:

Wolfram Language code: Subsets[f[a, b, c]]
Wolfram Language code: Subsets[a + b + c]

Applications  (7)

Find all ways to pick 3 elements from 4:

Wolfram Language code: Subsets[{1, 2, 3, 4}, {3}]
Wolfram Language code: Binomial[4, 3]

Draw lines between all pairs of points in an octagon:

Wolfram Language code: Graphics[Line[Subsets[Table[{Cos[2Pi i / 8], Sin[2 Pi i / 8]}, {i, 8}], {2}]]]

Construct an elementary symmetric polynomial:

Wolfram Language code: Total[Subsets[Times[a, b, c, d, e], {3}]]

All possible subsets of the divisors of 10:

Wolfram Language code: Subsets[Divisors[10]]

Find integers that have exactly 3 nonzero binary digits:

Wolfram Language code: Subsets[{1, 2, 4, 8, 16}, {3}]
Wolfram Language code: Total /@ %

Join all possible pairs of 20 random points in 3D:

Wolfram Language code: Graphics3D[Line /@ Subsets[RandomReal[100, {20, 3}], {2}]]

Draw lines between all pairs of points in a cube:

Wolfram Language code: Graphics3D[Line[Subsets[Tuples[{1, 0}, 3], {2}]]]

Properties & Relations  (3)

Subsets picks out first the elements that appear first in the input:

Wolfram Language code: Subsets[{c, b, a}]

Different occurrences of the same element are treated as distinct:

Wolfram Language code: Subsets[{a, b, b, b}]

Tuples gives all possible combinations and reorderings of elements:

Wolfram Language code: Tuples[{a, b, c}, 2]
Wolfram Language code: Subsets[{a, b, c}, {2}]

Possible Issues  (1)

Subsets[a,b,c]===Take[Subsets[a,b],c] only when all requested items are present:

Wolfram Language code: Subsets[Range[4], All, 8]
Wolfram Language code: Take[Subsets[Range[4], All], 8]

When items requested by c are missing, the ones present are returned:

Wolfram Language code: Subsets[Range[3], All, 10]

To suppress the warning message, use Off:

Wolfram Language code: Off[Subsets::take]
Wolfram Language code: Subsets[Range[3], All, 10]

Neat Examples  (2)

Polygons formed from all possible triples of random points:

Wolfram Language code: Graphics[{Opacity[0.01], Polygon /@ Subsets[RandomInteger[100, {20, 2}], {3}]}]
Wolfram Language code: Graphics3D[{Opacity[.1], Polygon /@ Subsets[RandomInteger[100, {10, 3}], {3}]}]

Draw all possible 3D triangular polygons that can be formed with points of coordinates , 0 or 1:

Wolfram Language code: Graphics3D[Polygon[Subsets[Tuples[{-1, 0, 1}, 3], {3}]]]

See Also

Tuples  IntegerDigits  Binomial  IntegerPartitions  SymmetricPolynomial  BellB  RandomSample  SubsetQ  Subsequences  Groupings  Permutations

Function Repository: LexicographicSubsets  SubsetsWithComplements  SelectSubsets  NthSubset  SubsetFromIndex  SubsetIndex  KSetPartitions  TupleIndex

Tech Notes

    ▪
  • Lists as Sets

Related Guides

    ▪
  • Discrete Mathematics
  • ▪
  • Operations on Sets
  • ▪
  • Finite Mathematics
  • ▪
  • Constructing Lists
  • ▪
  • Incrementals
  • ▪
  • Rearranging & Restructuring Lists
  • ▪
  • Math & Counting Operations on Lists

Related Links

  • An Elementary Introduction to the Wolfram Language : Rearranging Lists

History

Introduced in 2004 (5.1)

Wolfram Research (2004), Subsets, Wolfram Language function, https://reference.wolfram.com/language/ref/Subsets.html.

Text

Wolfram Research (2004), Subsets, Wolfram Language function, https://reference.wolfram.com/language/ref/Subsets.html.

CMS

Wolfram Language. 2004. "Subsets." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Subsets.html.

APA

Wolfram Language. (2004). Subsets. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Subsets.html

BibTeX

@misc{reference.wolfram_2026_subsets, author="Wolfram Research", title="{Subsets}", year="2004", howpublished="\url{https://reference.wolfram.com/language/ref/Subsets.html}", note=[Accessed: 01-September-2026]}

BibLaTeX

@online{reference.wolfram_2026_subsets, organization={Wolfram Research}, title={Subsets}, year={2004}, url={https://reference.wolfram.com/language/ref/Subsets.html}, note=[Accessed: 01-September-2026]}

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