
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


(setq
   Set1 (1 2 3 7 abc "def" (u v w))
   Set2 (2 3 5 hello (x y z))
   Set3 (3 hello (x y z)) )
 
 
# Element tests (any non-NIL value means "yes")
: (member "def" Set1)
-> ("def" (u v w))
 
: (member "def" Set2)
-> NIL
 
: (member '(x y z) Set2)
-> ((x y z))
 
 
# Union
: (uniq (append Set1 Set2))
-> (1 2 3 7 abc "def" (u v w) 5 hello (x y z))
 
 
# Intersection
: (sect Set1 Set2)
-> (2 3)
 
 
# Difference
: (diff Set1 Set2)
-> (1 7 abc "def" (u v w))
 
 
# Test for subset
: (not (diff Set1 Set2))
-> NIL  # Set1 is not a subset of Set2
 
: (not (diff Set3 Set2))
-> T  # Set3 is a subset of Set2
 
 
# Test for equality
: (= (sort (copy Set1)) (sort (copy Set2)))
-> NIL
 
: (= (sort (copy Set2)) (sort (copy Set2)))
-> T