Information on Subsets, Combinations and Product Spaces

This page gives a brief explanation of the input and output options available on the "Generate Subsets or Combinations" page. More extensive explanation is given on the following pages.

• Subsets
• Combinations
• Product Spaces

Subsets

A set is a collection of objects, and a subset is a sub-collection of those objects. The number of subsets of an n element set is 2ⁿ. In the Object Server, the set from which subsets are made is always the set [n] of the first n positive integers (i.e., [n] = {1,2,...,n}).

Required Input:

For both subset options, n is a required input.

Optional Input:

For subsets in lex order, k is an optional input which limits the subsets output to only those subsets containing no more than k elements.

Ignored Input:

For both subset options, the input values n₁,...,n₆ are ignored.

Bitstring Output:

Each subset S is listed as a bitstring b_n···b₂b₁ where b_i = 1 if i is in S and b_i = 0 if i is not in S.

List of Elements Output:

Each subset S (say of size p) is output as an ordered list {s₁,s₂,...,s_p} of its elements.

Bit Change Output:

If Gray code is selected under "What Type?", then the index of the bit that changes is output; furthermove, the bit that changes is shown in green (in the bitstring output).

Towers of Hanoi Output:

Ignored unless Gray code selected. Further explanation on the subset info page.

Combinations

Required Input:

For all combinations options, n and k are required inputs.

Optional Input:

None.

Ignored Input:

For all combinations options, the values n₁,...,n₆ are ignored.

Bitstring Output:

Each subset S is listed as a bitstring b_n···b₂b₁ where b_i = 1 if i is in S and b_i = 0 if i is not in S.

List of Elements Output:

Each subset S is output as an ordered list {s₁,s₂,...,s_p} of its elements.

Transposition Output:

If either of the two transposition methods is selected under "What Type?", then the indices of the two bits that are transposed are output; furthermove, the transposed bits are shown in green (in the bitstring output).

Towers of Hanoi Output:

Ignored.

Product Spaces

Required Input:

At least one of the values n₁,...,n₆.

Optional Input:

None.

Ignored Input:

The values n and k.

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