
Data Structure

This illustrates a data structure, a means of storing data within a program.

You may see other such structures in the Data Structures category.
A   set  is a collection of elements, without duplicates and without order.



Task

Show each of these set operations:

 Set creation
 Test m ∈ S -- "m is an element in set S"
 A ∪ B -- union; a set of all elements either in set A or in set B.
 A ∩ B -- intersection; a set of all elements in both set A and set B.
 A ∖ B -- difference; a set of all elements in set A, except those in set B.
 A ⊆ B -- subset; true if every element in set A is also in set B.
 A = B -- equality; true if every element of set A is in set B and vice versa.


As an option, show some other set operations.

(If A ⊆ B, but A ≠ B, then A is called a true or proper subset of B, written A ⊂ B or A ⊊ B.)

As another option, show how to modify a mutable set.



One might implement a set using an associative array (with set elements as array keys and some dummy value as the values).

One might also implement a set with a binary search tree, or with a hash table, or with an ordered array of binary bits (operated on with bit-wise binary operators).

The basic test, m ∈ S, is O(n) with a sequential list of elements, O(log n) with a balanced binary search tree, or (O(1) average-case, O(n) worst case) with a hash table.



See also

 Array
 Associative array: Creation, Iteration
 Collections
 Compound data type
 Doubly-linked list: Definition, Element definition, Element insertion, List Traversal, Element Removal
 Linked list
 Queue: Definition, Usage
 Set
 Singly-linked list: Element definition, Element insertion, List Traversal, Element Removal
 Stack


Prelude> import Data.Set
Prelude Data.Set> empty :: Set Integer -- Empty set
fromList []
Prelude Data.Set> let s1 = fromList [1,2,3,4,3] -- Convert list into set
Prelude Data.Set> s1
fromList [1,2,3,4]
Prelude Data.Set> let s2 = fromList [3,4,5,6] 
Prelude Data.Set> union s1 s2 -- Union
fromList [1,2,3,4,5,6]
Prelude Data.Set> intersection s1 s2 -- Intersection
fromList [3,4]
Prelude Data.Set> s1 \\ s2 -- Difference
fromList [1,2]
Prelude Data.Set> s1 `isSubsetOf` s1 -- Subset
True
Prelude Data.Set> fromList [3,1] `isSubsetOf` s1
True
Prelude Data.Set> s1 `isProperSubsetOf` s1 -- Proper subset
False
Prelude Data.Set> fromList [3,1] `isProperSubsetOf` s1
True
Prelude Data.Set> fromList [3,2,4,1] == s1 -- Equality
True
Prelude Data.Set> s1 == s2
False
Prelude Data.Set> 2 `member` s1 -- Membership
True
Prelude Data.Set> 10 `notMember` s1
True
Prelude Data.Set> size s1 -- Cardinality
4
Prelude Data.Set> insert 99 s1 -- Create a new set by inserting
fromList [1,2,3,4,99]
Prelude Data.Set> delete 3 s1 -- Create a new set by deleting
fromList [1,2,4]