# set

(data structure)

**Definition:**
An unordered collection of values where each value occurs at most once. A group of elements with three properties: (1) all elements belong to a *universe*, (2) either each element is a *member* of the set or it is not, and (3) the elements are unordered.

**Formal Definition:** As an *abstract data type*, a set has a single query function, isIn(v, S), which tells whether an element is a member of the set or not, and two modifier functions, add(v, S) and remove(v, S). These may be defined with *axiomatic semantics* as follows.

- new() returns a set
- isIn(v, new()) = false
- isIn(v, add(v, S)) = true
- isIn(v, add(u, S)) = isIn(v , S) if v ≠ u
- remove(v, new()) = new()
- remove(v, add(v, S)) = remove(v, S)
- remove(v, add(u, S)) = add(u, remove(v, S)) if v ≠ u

where S is a set and u and v are elements. The *predicate* isEmpty(S) may be defined with the following additional axioms.

- isEmpty(new()) = true
- isEmpty(add(v, S)) = false

**Generalization** (I am a kind of ...)

*bag*, *abstract data type*.

**Specialization** (... is a kind of me.)

*poset* suggested by Norman T. Thornton 25 January 2016.

**See also**
*intersection*, *union*, *complement*, *difference*, *list*, *set cover*.

*Note:
Sets may be implemented with an **array* of bits. A set that is many ranges may be efficiently implemented as an *inversion list*.

Authors: PR,PEB

## Implementation

(C++ and Pascal).

Go to the
Dictionary of Algorithms and Data
Structures home page.

If you have suggestions, corrections, or comments, please get in touch
with Paul Black.

Entry modified 25 March 2016.

HTML page formatted Fri Feb 23 10:06:08 2018.

Cite this as:

Patrick Rodgers and Paul E. Black, "set", in
*Dictionary of Algorithms and Data Structures* [online], Vreda Pieterse and Paul E. Black, eds. 25 March 2016. (accessed TODAY)
Available from: https://www.nist.gov/dads/HTML/set.html