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| ---- == 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}} |
Sets
Sets are a collection of members.
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. This can be formally expressed as ∀ a ((a ∈ A) -> (a ∈ B)).
This gives 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.
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:
