more on this theme
|
more from this text
Single Idea 9711
[filed under theme 4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
]
Full Idea
A relation is 'reflexive' on a set if every member of the set bears the relation to itself.
Gist of Idea
A relation is 'reflexive' on a set if every member bears the relation to itself
Source
Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0)
Book Ref
Enderton,Herbert B.: 'A Mathematical Introduction to Logic' [Academic Press 2001], p.5
The
38 ideas
from Herbert B. Enderton
13200
|
Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ
[Enderton]
|
13199
|
The empty set may look pointless, but many sets can be constructed from it
[Enderton]
|
13201
|
∈ says the whole set is in the other; ⊆ says the members of the subset are in the other
[Enderton]
|
13202
|
Fraenkel added Replacement, to give a theory of ordinal numbers
[Enderton]
|
13203
|
The singleton is defined using the pairing axiom (as {x,x})
[Enderton]
|
13204
|
The 'ordered pair' <x,y> is defined to be {{x}, {x,y}}
[Enderton]
|
13205
|
We can only define functions if Choice tells us which items are involved
[Enderton]
|
13206
|
A 'linear or total ordering' must be transitive and satisfy trichotomy
[Enderton]
|
9718
|
Validity is either semantic (what preserves truth), or proof-theoretic (following procedures)
[Enderton]
|
9719
|
A proof theory is 'sound' if its valid inferences entail semantic validity
[Enderton]
|
9720
|
A proof theory is 'complete' if semantically valid inferences entail proof-theoretic validity
[Enderton]
|
9724
|
Until the 1960s the only semantics was truth-tables
[Enderton]
|
9994
|
A truth assignment to the components of a wff 'satisfy' it if the wff is then True
[Enderton]
|
9721
|
A logical truth or tautology is a logical consequence of the empty set
[Enderton]
|
9723
|
Sentences with 'if' are only conditionals if they can read as A-implies-B
[Enderton]
|
9996
|
Expressions are 'decidable' if inclusion in them (or not) can be proved
[Enderton]
|
9722
|
Inference not from content, but from the fact that it was said, is 'conversational implicature'
[Enderton]
|
9997
|
For a reasonable language, the set of valid wff's can always be enumerated
[Enderton]
|
9995
|
Proof in finite subsets is sufficient for proof in an infinite set
[Enderton]
|
9715
|
An 'equivalence relation' is a reflexive, symmetric and transitive binary relation
[Enderton]
|
9716
|
We 'partition' a set into distinct subsets, according to each relation on its objects
[Enderton]
|
9717
|
A set is 'dominated' by another if a one-to-one function maps the first set into a subset of the second
[Enderton]
|
9713
|
A relation is 'transitive' if it can be carried over from two ordered pairs to a third
[Enderton]
|
9714
|
A relation satisfies 'trichotomy' if all pairs are either relations, or contain identical objects
[Enderton]
|
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]
|
9699
|
The 'powerset' of a set is all the subsets of a given set
[Enderton]
|
9700
|
Two sets are 'disjoint' iff their intersection is empty
[Enderton]
|
9702
|
A 'domain' of a relation is the set of members of ordered pairs in the relation
[Enderton]
|
9701
|
A 'relation' is a set of ordered pairs
[Enderton]
|
9706
|
A 'function' is a relation in which each object is related to just one other object
[Enderton]
|
9708
|
A function 'maps A into B' if the relating things are set A, and the things related to are all in B
[Enderton]
|
9709
|
A function 'maps A onto B' if the relating things are set A, and the things related to are set B
[Enderton]
|
9711
|
A relation is 'reflexive' on a set if every member bears the relation to itself
[Enderton]
|
9712
|
A relation is 'symmetric' on a set if every ordered pair has the relation in both directions
[Enderton]
|