# Module `Set.Make`

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

### Parameters

• `Ord : OrderedType`

### Signature

`type elt`` = Ord.t`

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 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