= Sets =

'''Sets''' are a collection of members.

<<TableOfContents>>

----



== Description ==

A '''set''' is a collection of members. Generally, sets are defined and notated like ''{a,b,c}''.

A set can also be defined by a condition. For example, ''{x ∈ R | x > 0}'' should be read as the set of all real values greater than 0.

The empty set is notated Ø.



=== Commonly Used Sets ===

The commonly re-used notations for basic sets of numbers are:
 * Natural numbers (''N'')
 * Integers (''Z'')
 * Rational numbers (''Q'')
 * Real numbers (''R'')

----



== Logic ==



=== Membership ===

''a ∈ A'' means ''a'' is a member of ''A''. Conversely, ''a ∉ A'' means ''a'' is not a member of ''A''.



=== Subsets ===

''A ⊆ B'' means that ''A'' is a '''subset''' of ''B''. ''∀ a ((a ∈ A) -> (a ∈ B))''.

This leads to a test for set equality: if ''A ⊆ B'' and ''B ⊆ A''.

If ''A ⊆ B'' and ''A != B'', then ''A'' is a '''proper subset''' of ''B''. This is notated as either ''A ⊂ B'' or ''A ⊊ B''.

The reverse [[Analysis/Relations|relation]] could be notated as ''B ⊇ A'', ''B ⊃ A'', and so on.



=== Complements ===

The '''complement''' of ''A'' contains all elements that are not members of ''A''. This is usually notated as ''A^c^''.

Taking the complement is an involution: ''(A^c^)^c^ = A''.

----



== Operations ==

The '''union''' of two sets contains all members of both. ''A ⋃ B = {x | x ∈ A or x ∈ B}''.

The union of all subsets ''A,,i,,'' can be expressed as:

{{attachment:union.svg}}

The '''intersection''' of two sets contains all members that are common between the two. ''A ⋂ B = {x | x ∈ A and x ∈ B}''.

The intersection of all subsets ''A,,i,,'' can be expressed as:

{{attachment:intersection.svg}}

A pair of sets are '''disjoint''' if there is no intersection, i.e., ''A ⋂ B = ∅''.

The '''set difference''' of ''A'' with respect to ''B'' contains all members in ''A'' that are not in ''B''. ''A \ B = {x ∈ A | x ∉ B}''.

The '''Cartesian product''' of ''A'' and ''B'' is the set of all pairs ''(x,y)''. ''A × B = {(x,y) | x ∈ A and y ∈ B}''.



=== Properties ===

Let ''U'' be the universe of set ''A''. It is always true that ''A ⋂ A^c^ = ∅'' and ''A ⋃ A^c^ = U''. Furthermore, ''∅^c^ = U'' and ''U^c^ = ∅''.

'''De Morgan's laws''' prove that:
 * ''(A ⋂ B)^c^ = A^c^ ⋃ B^c^''
 * ''(A ⋃ B)^c^ = A^c^ ⋂ B^c^''

For set differences, ∅ has an identity property. ''A \ ∅ = A''.

Lastly, note that ''A^c^ \ B^c^ = B \ A''.



----
CategoryRicottone