# Module `Set.Make`

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

### Parameters

### Signature

`type elt`

`= Ord.t`

The type of the set elements.

`val empty : t`

The empty set.

`val is_empty : t -> bool`

Test whether a set is empty or not.

`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 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 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 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 a`

computes`(f xN ... (f x2 (f x1 a))...)`

, where`x1 ... xN`

are the elements of`s`

, in increasing order.

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

`for_all p s`

checks if all elements of the set satisfy the predicate`p`

.

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

`exists p s`

checks if at least one element of the set satisfies the predicate`p`

.

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

`filter p s`

returns the set of all elements in`s`

that satisfy predicate`p`

. If`p`

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

`partition p s`

returns a pair of sets`(s1, s2)`

, where`s1`

is the set of all the elements of`s`

that satisfy the predicate`p`

, and`s2`

is the set of all the elements of`s`

that do not satisfy`p`

.

`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 Stdlib.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 Stdlib.Seq.t`

Iterate on the whole set, in ascending order

- since
- 4.07

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

Add the given elements to the set, in order.

- since
- 4.07

`val of_seq : elt Stdlib.Seq.t -> t`

Build a set from the given bindings

- since
- 4.07