# Functor Set.Make

`module Make: `functor (``Ord`` : ``OrderedType``) -> ``S``  with type elt = Ord.t``

Functor building an implementation of the set structure given a totally ordered type.

Parameters:
 `Ord` : `OrderedType`

`type elt `

The type of the set elements.

`type t `

The type of sets.

`val empty : `t``

The empty set.

`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 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 4.08.0
`val diff : `t -> t -> t``

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

`val compare : `t -> t -> int``

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

`val equal : `t -> t -> bool``

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

`val subset : `t -> t -> bool``

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

`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 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 4.04.0
`val fold : `(elt -> 'a -> 'a) -> t -> 'a -> 'a``

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

`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`.

`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 4.11.0
`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 cardinal : `t -> int``

Return the number of elements of a set.

`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
`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`.

`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 4.01.0
`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
`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 4.02.0

## Iterators

`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