Subsets—Wolfram Language Documentation (original) (raw)

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
  • 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 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 allclose all

Basic Examples (2)

All possible subsets (power set):

All possible subsets containing up to 2 elements:

Subsets containing exactly 2 elements:

Scope (4)

The first 5 subsets containing 3 elements:

All subsets with even length:

The 69381 subset:

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

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

Generalizations & Extensions (1)

Applications (7)

Find all ways to pick 3 elements from 4:

Draw lines between all pairs of points in an octagon:

Construct an elementary symmetric polynomial:

All possible subsets of the divisors of 10:

Find integers that have exactly 3 nonzero binary digits:

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

Draw lines between all pairs of points in a cube:

Properties & Relations (3)

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

Different occurrences of the same element are treated as distinct:

Tuples gives all possible combinations and reorderings of elements:

Possible Issues (1)

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

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

To suppress the warning message, use Off:

Neat Examples (2)

Polygons formed from all possible triples of random points:

Related Guides

▪

• Discrete Mathematics ▪
• Operations on Sets ▪
• Finite Mathematics ▪
• Constructing Lists ▪
• Rearranging & Restructuring Lists ▪
• Math & Counting Operations on 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_2025_subsets, author="Wolfram Research", title="{Subsets}", year="2004", howpublished="\url{https://reference.wolfram.com/language/ref/Subsets.html}", note=[Accessed: 17-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_subsets, organization={Wolfram Research}, title={Subsets}, year={2004}, url={https://reference.wolfram.com/language/ref/Subsets.html}, note=[Accessed: 17-June-2025 ]}