
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


>>> s1, s2 = {1, 2, 3, 4}, {3, 4, 5, 6}
>>> s1 | s2 # Union
{1, 2, 3, 4, 5, 6}
>>> s1 & s2 # Intersection
{3, 4}
>>> s1 - s2 # Difference
{1, 2}
>>> s1 < s1 # True subset
False
>>> {3, 1} < s1 # True subset
True
>>> s1 <= s1 # Subset
True
>>> {3, 1} <= s1 # Subset
True
>>> {3, 2, 4, 1} == s1 # Equality
True
>>> s1 == s2 # Equality
False
>>> 2 in s1 # Membership
True
>>> 10 not in s1 # Non-membership
True
>>> {1, 2, 3, 4, 5} > s1 # True superset
True
>>> {1, 2, 3, 4} > s1 # True superset
False
>>> {1, 2, 3, 4} >= s1 # Superset
True
>>> s1 ^ s2 # Symmetric difference
{1, 2, 5, 6}
>>> len(s1) # Cardinality
4
>>> s1.add(99) # Mutability
>>> s1
{99, 1, 2, 3, 4}
>>> s1.discard(99) # Mutability
>>> s1
{1, 2, 3, 4}
>>> s1 |= s2 # Mutability
>>> s1
{1, 2, 3, 4, 5, 6}
>>> s1 -= s2 # Mutability
>>> s1
{1, 2}
>>> s1 ^= s2 # Mutability
>>> s1
{1, 2, 3, 4, 5, 6}
>>> 