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| ''A ⊆ B'' means that ''A'' is a '''subset''' of ''B''. This can be formally expressed as ''∀ a ((a ∈ A) -> (a ∈ B))''. | ''A ⊆ B'' means that ''A'' is a '''subset''' of ''B''. ''∀ a ((a ∈ A) -> (a ∈ B))''. |
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| This gives a test for set equality: if ''A ⊆ B'' and ''B ⊆ A''. | This leads to a test for set equality: if ''A ⊆ B'' and ''B ⊆ A''. |
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| The reverse relation could be notated as ''B ⊇ A'', ''B ⊃ A'', and so on. | 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''. |
Sets
Sets are a collection of members.
Contents
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 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 Ac.
Taking the complement is an involution: (Ac)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 Ai can be expressed as:
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 Ai can be expressed as:
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 ⋂ Ac = ∅ and A ⋃ Ac = U. Furthermore, ∅c = U and Uc = ∅.
De Morgan's laws prove that:
(A ⋂ B)c = Ac ⋃ Bc
(A ⋃ B)c = Ac ⋂ Bc
For set differences, ∅ has an identity property. A \ ∅ = A.
Lastly, note that Ac \ Bc = B \ A.