
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


 
> S := { 2, 3, 5, 7, 11, Pi, "foo", { 2/3, 3/4, 4/5 } };
           S := {2, 3, 5, 7, 11, "foo", Pi, {2/3, 3/4, 4/5}}
 
> type( S, set );
                                  true
 
> Pi in S;
          Pi  in  {2, 3, 5, 7, 11, "foo", Pi, {2/3, 3/4, 4/5}}
 
> if Pi in S then print( yes ) else print( no ) end:
                                  yes
 
> member( Pi, S );
                                  true
 
> if 4 in S then print( yes ) else print( no ) end:
                                   no
 
> evalb( { 2/3, 3/4, 4/5 } in S );
                                  true
 
> { a, b, c } union { 1, 2, 3 };
                           {1, 2, 3, a, b, c}
 
> { a, b, c } intersect { b, c, d };
                                 {b, c}
 
> { a, b, c } minus { b, c, d };
                                  {a}
 
> { a, b } subset { a, b, c };
                                  true
 
> { a, d } subset { a, b, c };
                                 false
 
> evalb( { 1, 2, 3 } = { 1, 2, 3 } );
                                  true
 
> evalb( { 1, 2, 3 } = { 1, 2, 4 } );
                                 false
 