## 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

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 bn···b2b1 where bi = 1 if i is in S and bi = 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,...,sp} 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 bn···b2b1 where bi = 1 if i is in S and bi = 0 if i is not in S.
List of Elements Output:
Each subset S is output as an ordered list {s1,s2,...,sp} 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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