# Subsets¶

class sympy.combinatorics.subsets.Subset(subset, superset)[source]

Represents a basic subset object.

Explanation

We generate subsets using essentially two techniques, binary enumeration and lexicographic enumeration. The Subset class takes two arguments, the first one describes the initial subset to consider and the second describes the superset.

Examples

```>>> from sympy.combinatorics import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.next_binary().subset
['b']
>>> a.prev_binary().subset
['c']
```
classmethod bitlist_from_subset(subset, superset)[source]

Gets the bitlist corresponding to a subset.

Examples

```>>> from sympy.combinatorics import Subset
>>> Subset.bitlist_from_subset(['c', 'd'], ['a', 'b', 'c', 'd'])
'0011'
```
property cardinality

Returns the number of all possible subsets.

Examples

```>>> from sympy.combinatorics import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.cardinality
16
```
iterate_binary(k)[source]

This is a helper function. It iterates over the binary subsets by `k` steps. This variable can be both positive or negative.

Examples

```>>> from sympy.combinatorics import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.iterate_binary(-2).subset
['d']
>>> a = Subset(['a', 'b', 'c'], ['a', 'b', 'c', 'd'])
>>> a.iterate_binary(2).subset
[]
```
iterate_graycode(k)[source]

Helper function used for prev_gray and next_gray. It performs `k` step overs to get the respective Gray codes.

Examples

```>>> from sympy.combinatorics import Subset
>>> a = Subset([1, 2, 3], [1, 2, 3, 4])
>>> a.iterate_graycode(3).subset
[1, 4]
>>> a.iterate_graycode(-2).subset
[1, 2, 4]
```

next_binary()[source]

Generates the next binary ordered subset.

Examples

```>>> from sympy.combinatorics import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.next_binary().subset
['b']
>>> a = Subset(['a', 'b', 'c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.next_binary().subset
[]
```
next_gray()[source]

Generates the next Gray code ordered subset.

Examples

```>>> from sympy.combinatorics import Subset
>>> a = Subset([1, 2, 3], [1, 2, 3, 4])
>>> a.next_gray().subset
[1, 3]
```
next_lexicographic()[source]

Generates the next lexicographically ordered subset.

Examples

```>>> from sympy.combinatorics import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.next_lexicographic().subset
['d']
>>> a = Subset(['d'], ['a', 'b', 'c', 'd'])
>>> a.next_lexicographic().subset
[]
```
prev_binary()[source]

Generates the previous binary ordered subset.

Examples

```>>> from sympy.combinatorics import Subset
>>> a = Subset([], ['a', 'b', 'c', 'd'])
>>> a.prev_binary().subset
['a', 'b', 'c', 'd']
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.prev_binary().subset
['c']
```
prev_gray()[source]

Generates the previous Gray code ordered subset.

Examples

```>>> from sympy.combinatorics import Subset
>>> a = Subset([2, 3, 4], [1, 2, 3, 4, 5])
>>> a.prev_gray().subset
[2, 3, 4, 5]
```
prev_lexicographic()[source]

Generates the previous lexicographically ordered subset.

Examples

```>>> from sympy.combinatorics import Subset
>>> a = Subset([], ['a', 'b', 'c', 'd'])
>>> a.prev_lexicographic().subset
['d']
>>> a = Subset(['c','d'], ['a', 'b', 'c', 'd'])
>>> a.prev_lexicographic().subset
['c']
```
property rank_binary

Computes the binary ordered rank.

Examples

```>>> from sympy.combinatorics import Subset
>>> a = Subset([], ['a','b','c','d'])
>>> a.rank_binary
0
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.rank_binary
3
```
property rank_gray

Computes the Gray code ranking of the subset.

Examples

```>>> from sympy.combinatorics import Subset
>>> a = Subset(['c','d'], ['a','b','c','d'])
>>> a.rank_gray
2
>>> a = Subset([2, 4, 5], [1, 2, 3, 4, 5, 6])
>>> a.rank_gray
27
```
property rank_lexicographic

Computes the lexicographic ranking of the subset.

Examples

```>>> from sympy.combinatorics import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.rank_lexicographic
14
>>> a = Subset([2, 4, 5], [1, 2, 3, 4, 5, 6])
>>> a.rank_lexicographic
43
```
property size

Gets the size of the subset.

Examples

```>>> from sympy.combinatorics import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.size
2
```
property subset

Gets the subset represented by the current instance.

Examples

```>>> from sympy.combinatorics import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.subset
['c', 'd']
```
classmethod subset_from_bitlist(super_set, bitlist)[source]

Gets the subset defined by the bitlist.

Examples

```>>> from sympy.combinatorics import Subset
>>> Subset.subset_from_bitlist(['a', 'b', 'c', 'd'], '0011').subset
['c', 'd']
```
classmethod subset_indices(subset, superset)[source]

Return indices of subset in superset in a list; the list is empty if all elements of `subset` are not in `superset`.

Examples

```>>> from sympy.combinatorics import Subset
>>> superset = [1, 3, 2, 5, 4]
>>> Subset.subset_indices([3, 2, 1], superset)
[1, 2, 0]
>>> Subset.subset_indices([1, 6], superset)
[]
>>> Subset.subset_indices([], superset)
[]
```
property superset

Gets the superset of the subset.

Examples

```>>> from sympy.combinatorics import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.superset
['a', 'b', 'c', 'd']
```
property superset_size

Returns the size of the superset.

Examples

```>>> from sympy.combinatorics import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.superset_size
4
```
classmethod unrank_binary(rank, superset)[source]

Gets the binary ordered subset of the specified rank.

Examples

```>>> from sympy.combinatorics import Subset
>>> Subset.unrank_binary(4, ['a', 'b', 'c', 'd']).subset
['b']
```
classmethod unrank_gray(rank, superset)[source]

Gets the Gray code ordered subset of the specified rank.

Examples

```>>> from sympy.combinatorics import Subset
>>> Subset.unrank_gray(4, ['a', 'b', 'c']).subset
['a', 'b']
>>> Subset.unrank_gray(0, ['a', 'b', 'c']).subset
[]
```
subsets.ksubsets(k)[source]

Finds the subsets of size `k` in lexicographic order.

This uses the itertools generator.

Examples

```>>> from sympy.combinatorics.subsets import ksubsets
>>> list(ksubsets([1, 2, 3], 2))
[(1, 2), (1, 3), (2, 3)]
>>> list(ksubsets([1, 2, 3, 4, 5], 2))
[(1, 2), (1, 3), (1, 4), (1, 5), (2, 3), (2, 4),     (2, 5), (3, 4), (3, 5), (4, 5)]
```