# Subsets of a Finite Set

## `Subsets(S): SetEnum -> SetEnum`

The set of all subsets of the set $S$.

## `Subsets(S, k): SetEnum, RngIntElt -> SetEnum`

The set of subsets of the set $S$ of size $k$. If $k$ is larger than the cardinality of $S$ then the result will be empty.

## `Multisets(S, k): SetEnum, RngIntElt -> SetEnum`

The set of multisets consisting of $k$ not necessarily distinct elements of the set $S$.

## `Subsequences(S, k): SetEnum, RngIntElt -> SetEnum`

The set of sequences of length $k$ with elements from the set $S$.

## `Permutations(S): SetEnum -> SetEnum;`

The set of permutations (stored as sequences) of the elements of the set $S$.

## `Permutations(S, k): SetEnum, RngIntElt -> SetEnum;`

The set of permutations (stored as sequences) of each of the subsets of the set $S$ of cardinality $k$.

## `Example: Odd Graph (ex-676132)`

The use of Subsets is illustrated in the construction of the Petersen graph as the third Odd Graph. The $n$th Odd Graph has its vertices in correspondence with the $(n-1)$-element subsets of $\lbrace 1 \ldots 2n-1 \rbrace$, and an edge between two vertices if and only if their corresponding sets have empty intersection.

```magma
> V := Subsets( {1 .. 2*n-1}, n-1) where n is 3;
> V;
{
    { 1, 5 },
    { 2, 5 },
    { 1, 3 },
    { 1, 4 },
    { 2, 4 },
    { 3, 5 },
    { 2, 3 },
    { 1, 2 },
    { 3, 4 },
    { 4, 5 }
}
> E := { {u, v} : u,v in V | IsDisjoint(u, v) };
> Petersen := Graph< V | E >;

```