Power Set by Subset Size

Formula and method

For every element, independently choose whether to include it. This gives 2ⁿ subsets of an n-element set. Grouping them by their number of elements gives the binomial counts C(n,k).

|P(S)|=2ⁿ; the number of k-element subsets is C(n,k).

Worked example

Enter these known values and leave the other values blank.

Assumptions and limitations

• Enter one through ten distinct numeric elements. Blanks are omitted. Zero is a valid element, while −0 and 0 denote the same element.
• Elements may contain at most seven significant digits and have magnitude from 1e-1000 through 1e1000, or be zero. Longer values reject explicitly rather than being rounded into duplicate elements.
• Equivalent spellings such as 1 and 1.0, or 1e3 and 1000, are duplicates. Remove repeated values to obtain a set.
• The calculator lists the empty subset and the complete subset exactly once, preserves entered element order inside subsets, and groups by increasing cardinality.
• At most 1024 subsets and 131072 total plain-text characters are generated. The source has no dedicated empty-set input mode; with every control blank it asks for an element. Mathematically P(∅)={∅}.
• Inputs accept decimal or e notation and bounded arithmetic expressions only when each intermediate and final value can be verified exactly as a finite decimal. Repeating fractions, functions, symbols and expressions that lose digits are rejected with guidance. Enter their finite decimal value instead; the calculator-specific magnitude and precision limits still apply.

Common questions

References

• Captured ten-element controls and subset grouping

Translate the captured ten-element power-set callback into bounded local exact canonical-element enumeration. Preserve all ten progressive controls, original hidden text-choice datasets, duplicate/precision guidance, empty-input prompt, empty subset, full subset, stable cardinality grouping and binomial counts. Replace its seven-digit rounding/character-length conflation with explicit seven-significant-digit validation; reject duplicates rather than silently merging them. Adapt the two valueMarkdown result exports to declared bounded plain-text paragraphs only, with no Markdown/HTML execution. Bind this complete power-set source/schema to at most ten distinct finite numeric elements, each zero or magnitude1e-1000 through1e1000 and at most seven significant digits. Input display preserves accepted canonical digits. The1024-subset maximum and22-character canonical element bound prove131072total text characters; retain at most eight immutable cached summaries. Preserve all eleven possible cardinality groups as separate plain paragraphs, hide groups above the actual set size, and retain both endpoints exactly once. Exact-sign pair conditions reject repeated equivalent elements at every scale. Empty controls remain absent; the calculator retains its captured at-least-one-element prompt instead of inventing an empty-set mode.

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