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A set is '''open''' if every members |
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| '''Openness''' of a set can be determined in either a [[Analysis/TopologicalSpace|topological space]] or a [[Analysis/MetricSpace|metric space]]. The core idea is that a set is closed if it contains all of its [[Analysis/LimitPoint|limit points]], and open if it contains none of them. |
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The empty set is notated Ø. |
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 size of a set is characterized by cardinality: they can be countable (i.e., there are n members), countably infinite (i.e., the set can be mapped to the natural numbers), or uncountable.
A set may or may not be ordered.
If a set is closed on addition and multiplication, it is actually then a field.
Openness of a set can be determined in either a topological space or a metric space. The core idea is that a set is closed if it contains all of its limit points, and open if it contains none of them.
Commonly Used Sets
The empty set is notated Ø.
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.