more from this thinker
|
more from this text
Single Idea 13443
[filed under theme 4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / a. Symbols of ST
]
Full Idea
∈ relates across layers (Plato is a member of his unit set and the set of people), while ⊆ relates within layers (the singleton of Plato is a subset of the set of people). This distinction only became clear in the 19th century.
Gist of Idea
∈ relates across layers, while ⊆ relates within layers
Source
William D. Hart (The Evolution of Logic [2010], 1)
Book Ref
Hart,W.D.: 'The Evolution of Logic' [CUP 2010], p.5
A Reaction
Getting these two clear may be the most important distinction needed to understand how set theory works.
Related Ideas
Idea 13201
∈ says the whole set is in the other; ⊆ says the members of the subset are in the other [Enderton]
Idea 12337
There is 'transivity' iff membership ∈ also means inclusion ⊆ [Badiou]
Idea 15500
Classes divide into subclasses in many ways, but into members in only one way [Lewis]
Idea 15499
A subclass of a subclass is itself a subclass; a member of a member is not in general a member [Lewis]
The
19 ideas
with the same theme
[symbols which are distinctive of set theory]:
|
9703
|
'dom R' indicates the 'domain' of objects having a relation
[Enderton]
|
|
9705
|
'fld R' indicates the 'field' of all objects in the relation
[Enderton]
|
|
9704
|
'ran R' indicates the 'range' of objects being related to
[Enderton]
|
|
9710
|
We write F:A→B to indicate that A maps into B (the output of F on A is in B)
[Enderton]
|
|
9707
|
'F(x)' is the unique value which F assumes for a value of x
[Enderton]
|
|
13443
|
∈ relates across layers, while ⊆ relates within layers
[Hart,WD]
|
|
9697
|
X1 x X2 x X3... x Xn indicates the 'cartesian product' of those sets
[Priest,G]
|
|
9685
|
<a,b&62; is a set whose members occur in the order shown
[Priest,G]
|
|
9675
|
a ∈ X says a is an object in set X; a ∉ X says a is not in X
[Priest,G]
|
|
9674
|
{x; A(x)} is a set of objects satisfying the condition A(x)
[Priest,G]
|
|
9673
|
{a1, a2, ...an} indicates that a set comprising just those objects
[Priest,G]
|
|
9677
|
Φ indicates the empty set, which has no members
[Priest,G]
|
|
9676
|
{a} is the 'singleton' set of a (not the object a itself)
[Priest,G]
|
|
9679
|
X⊂Y means set X is a 'proper subset' of set Y
[Priest,G]
|
|
9678
|
X⊆Y means set X is a 'subset' of set Y
[Priest,G]
|
|
9681
|
X = Y means the set X equals the set Y
[Priest,G]
|
|
9683
|
X ∩ Y indicates the 'intersection' of sets X and Y, the objects which are in both sets
[Priest,G]
|
|
9682
|
X∪Y indicates the 'union' of all the things in sets X and Y
[Priest,G]
|
|
9684
|
Y - X is the 'relative complement' of X with respect to Y; the things in Y that are not in X
[Priest,G]
|