# Module type Set.S

    module type S = sig .. end

Output signature of the functor `Set.Make`.

## Sets

    type elt

The type of the set elements.

    type t

The type of sets.

    val empty : t

The empty set.

    val add : elt -> t -> t

`add x s` returns a set containing all elements of `s`, plus `x`. If `x` was already in `s`, `s` is returned unchanged (the result of the function is then physically equal to `s`).

- Before 4.03 Physical equality was not ensured.

    val singleton : elt -> t

`singleton x` returns the one-element set containing only `x`.

    val remove : elt -> t -> t

`remove x s` returns a set containing all elements of `s`, except `x`. If `x` was not in `s`, `s` is returned unchanged (the result of the function is then physically equal to `s`).

- Before 4.03 Physical equality was not ensured.

    val union : t -> t -> t

Set union.

    val inter : t -> t -> t

Set intersection.

    val disjoint : t -> t -> bool

Test if two sets are disjoint.

- Since 4.08

    val diff : t -> t -> t

Set difference: `diff s1 s2` contains the elements of `s1` that are not in `s2`.

    val cardinal : t -> int

Return the number of elements of a set.

## Elements

    val elements : t -> elt list

Return the list of all elements of the given set. The returned list is sorted in increasing order with respect to the ordering `Ord.compare`, where `Ord` is the argument given to `Set.Make`.

    val min_elt : t -> elt

Return the smallest element of the given set (with respect to the `Ord.compare` ordering), or raise `Not_found` if the set is empty.

    val min_elt_opt : t -> elt option

Return the smallest element of the given set (with respect to the `Ord.compare` ordering), or `None` if the set is empty.

- Since 4.05

    val max_elt : t -> elt

Same as `Set.S.min_elt`, but returns the largest element of the given set.

    val max_elt_opt : t -> elt option

Same as `Set.S.min_elt_opt`, but returns the largest element of the given set.

- Since 4.05

    val choose : t -> elt

Return one element of the given set, or raise `Not_found` if the set is empty. Which element is chosen is unspecified, but equal elements will be chosen for equal sets.

    val choose_opt : t -> elt option

Return one element of the given set, or `None` if the set is empty. Which element is chosen is unspecified, but equal elements will be chosen for equal sets.

- Since 4.05

## Searching

    val find : elt -> t -> elt

`find x s` returns the element of `s` equal to `x` (according to `Ord.compare`), or raise `Not_found` if no such element exists.

- Since 4.01

    val find_opt : elt -> t -> elt option

`find_opt x s` returns the element of `s` equal to `x` (according to `Ord.compare`), or `None` if no such element exists.

- Since 4.05

    val find_first : (elt -> bool) -> t -> elt

`find_first f s`, where `f` is a monotonically increasing function, returns the lowest element `e` of `s` such that `f e`, or raises `Not_found` if no such element exists.

For example, `find_first (fun e -> Ord.compare e x >= 0) s` will return the first element `e` of `s` where `Ord.compare e x >= 0` (intuitively: `e >= x`), or raise `Not_found` if `x` is greater than any element of `s`.

- Since 4.05

    val find_first_opt : (elt -> bool) -> t -> elt option

`find_first_opt f s`, where `f` is a monotonically increasing function, returns an option containing the lowest element `e` of `s` such that `f e`, or `None` if no such element exists.

- Since 4.05

    val find_last : (elt -> bool) -> t -> elt

`find_last f s`, where `f` is a monotonically decreasing function, returns the highest element `e` of `s` such that `f e`, or raises `Not_found` if no such element exists.

- Since 4.05

    val find_last_opt : (elt -> bool) -> t -> elt option

`find_last_opt f s`, where `f` is a monotonically decreasing function, returns an option containing the highest element `e` of `s` such that `f e`, or `None` if no such element exists.

- Since 4.05

## Traversing

    val iter : (elt -> unit) -> t -> unit

`iter f s` applies `f` in turn to all elements of `s`. The elements of `s` are presented to `f` in increasing order with respect to the ordering over the type of the elements.

    val fold : (elt -> 'acc -> 'acc) -> t -> 'acc -> 'acc

`fold f s init` computes `(f xN ... (f x2 (f x1 init))...)`, where `x1 ... xN` are the elements of `s`, in increasing order.

## Transforming

    val map : (elt -> elt) -> t -> t

`map f s` is the set whose elements are `f a0`,`f a1`... `f
aN`, where `a0`,`a1`...`aN` are the elements of `s`.

The elements are passed to `f` in increasing order with respect to the ordering over the type of the elements.

If no element of `s` is changed by `f`, `s` is returned unchanged. (If each output of `f` is physically equal to its input, the returned set is physically equal to `s`.)

- Since 4.04

    val filter : (elt -> bool) -> t -> t

`filter f s` returns the set of all elements in `s` that satisfy predicate `f`. If `f` satisfies every element in `s`, `s` is returned unchanged (the result of the function is then physically equal to `s`).

- Before 4.03 Physical equality was not ensured.

    val filter_map : (elt -> elt option) -> t -> t

`filter_map f s` returns the set of all `v` such that `f x = Some v` for some element `x` of `s`.

For example,

    filter_map (fun n -> if n mod 2 = 0 then Some (n / 2) else None) s

is the set of halves of the even elements of `s`.

If no element of `s` is changed or dropped by `f` (if `f x = Some x` for each element `x`), then `s` is returned unchanged: the result of the function is then physically equal to `s`.

- Since 4.11

    val partition : (elt -> bool) -> t -> t * t

`partition f s` returns a pair of sets `(s1, s2)`, where `s1` is the set of all the elements of `s` that satisfy the predicate `f`, and `s2` is the set of all the elements of `s` that do not satisfy `f`.

    val split : elt -> t -> t * bool * t

`split x s` returns a triple `(l, present, r)`, where `l` is the set of elements of `s` that are strictly less than `x`; `r` is the set of elements of `s` that are strictly greater than `x`; `present` is `false` if `s` contains no element equal to `x`, or `true` if `s` contains an element equal to `x`.

## Predicates and comparisons

    val is_empty : t -> bool

Test whether a set is empty or not.

    val mem : elt -> t -> bool

`mem x s` tests whether `x` belongs to the set `s`.

    val equal : t -> t -> bool

`equal s1 s2` tests whether the sets `s1` and `s2` are equal, that is, contain equal elements.

    val compare : t -> t -> int

Total ordering between sets. Can be used as the ordering function for doing sets of sets.

    val subset : t -> t -> bool

`subset s1 s2` tests whether the set `s1` is a subset of the set `s2`.

    val for_all : (elt -> bool) -> t -> bool

`for_all f s` checks if all elements of the set satisfy the predicate `f`.

    val exists : (elt -> bool) -> t -> bool

`exists f s` checks if at least one element of the set satisfies the predicate `f`.

## Converting

    val to_list : t -> elt list

`to_list s` is `Set.S.elements` `s`.

- Since 5.1

    val of_list : elt list -> t

`of_list l` creates a set from a list of elements. This is usually more efficient than folding `add` over the list, except perhaps for lists with many duplicated elements.

- Since 4.02

    val to_seq_from : elt -> t -> elt Seq.t

`to_seq_from x s` iterates on a subset of the elements of `s` in ascending order, from `x` or above.

- Since 4.07

    val to_seq : t -> elt Seq.t

Iterate on the whole set, in ascending order

- Since 4.07

    val to_rev_seq : t -> elt Seq.t

Iterate on the whole set, in descending order

- Since 4.12

    val add_seq : elt Seq.t -> t -> t

Add the given elements to the set, in order.

- Since 4.07

    val of_seq : elt Seq.t -> t

Build a set from the given bindings

- Since 4.07