User:Prime.mover
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[edit] Contact Me
I'm on the local email as prime.mover@proofwiki.org so drop us a line when you want.
Or you can leave a message on my talk page.
But please answer this question first:
You might also catch me asking (but usually answering) questions on Math Help Forum which I sometimes haunt when ProofWiki's down.
[edit] Workload
I made a start on:
- Propositional Logic
- Naive Set Theory
- Function Theory
- Abstract Algebra (including dabbling in a little group theory)
- Real Analysis
- Metric Spaces
- Number Theory
- Currently battling through Mathematical Logic
[edit] What's THIS For ...!
Sandbox, In Progress, Later, Help Needed.
{{POTW Candidate}}
Prime.mover (talk) (contributions) (NUM)
[edit] URM Programs
| Line | Command | Comment | ||
|---|---|---|---|---|
| | |||
| | |||
| | |||
| |
...etc.
Let
be a URM program.
Let
be a normalized URM program.
Let
be the number of basic instructions in
.
Let
be the number of registers used by
.
Trace Table:
| Stage | Instruction |
|
|
|
|---|---|---|---|---|
|
|
|
|
|
|
|
|
|
|
...etc.
[edit] Useful constructs
Useful constructs for anyone to cut and paste:
Blackboard characters:
- Redirect User:Prime.mover#Barnstars
For example:
Let
be a probability space.
Let
be a discrete random variable on
.
Let
be the floor of
.
Let
be the ceiling of
.
Let
be a topological space.
Let
be a metric space.
Let
be an
-neighborhood in
.
Let
be a real number.
Let
be a power series about
.
Let
be a real function which is continuous on the closed interval
and differentiable on the open interval
.
Let
have a primitive
on
.
Let
be a convergent series in
.
Let
be the sequence of partial sums of
.
Let
be a sequence in
.
Let
be a Cauchy sequence.
Let
.
Let
as
.
Let
be a subsequence of
.
Let
be an
matrix.
Let
be a square matrix of order
.
Let
be the determinant of
.
Let
be the
matrix space over
.
Let
be a set.
Let
be the power set of the set
.
Let
be an algebraic structure or a semigroup.
Let
be a group whose identity is
.
Let
be a ring with unity whose zero is
and whose unity is
.
Let
be a division ring whose zero is
and whose unity is
.
Let
be the group generated by
.
Let
be a cyclic group.
Let
be an
-module.
Let
be a
-vector space.
Let
be a unitary
-module
whose dimension is finite.
Let
be the set of all linear transformations from
to
.
Let
be the set of all linear operators on
.
Let
be the matrix of
relative to
and
.
Let
be the set of polynomials in
over
.
Let
be the ring of polynomial forms in
over
.
Let
be the ring of polynomial functions over
.
Let
be the algebraic dual of
.
Let
be the algebraic dual of
.
Let
be the annihilator of
.
Let
be as defined in Evaluation Linear Transformation.
Let
be an ideal of
.
Let
be the quotient ring defined by
.
Let
be an integral domain or a principal ideal domain whose zero is
and whose unity is
.
Let
be a field whose zero is
and whose unity is
.
Let
be a quotient field of an integral domain
.
Let
be a totally ordered integral domain whose zero is
and whose unity is
.
Let
be a totally ordered set.
Let
be an ordered structure.
Let
be a naturally ordered semigroup.
Let
be a Naturally Ordered Semigroup with Product.
is the closed interval between
and
.
,
,
,
,
,
,
,
Let
be the set of integers modulo
.
Let
be the set of integers coprime to
in
.
Let
be the Additive Group of Integers.
Let
be the integral domain of integers.
Let
be the ring of integers modulo
.
Let
be the Additive Group of Integers Modulo
.
Let
be the set of integer multiples of
.
Let
be the principal ideal of
generated by
.
Let
be the characteristic of
.
The cardinality of a set
is written
.
Let
be a sequence in
.
Let
be the greatest common divisor of
and
.
Let
be the lowest common multiple of
and
.
Let
be the absolute value of
.
: "
is congruent to
modulo
."
is the residue class of
(modulo
).
Let
be the index of
in
.
Let
be the centralizer of
in
.
Let
be the normalizer of
in
.
Let
be the quotient group of
by
.
Let
be the center of
.
Let
.
Let
be the normalizer of
in
.
Let
be the index of
in
.
Let
denote the set of permutations on
letters.
Let
denote the symmetric group on
letters.
Let
be the set of elements fixed by
.
Matrix (square brackets):
Matrix (round brackets):
Let
be the orbit of
.
Let
be the stabilizer of
by
.
Let
be an
-algebraic structure.
[edit] Ordinary proofs
| ||||||
|
|
...etc.
[edit] Iff Proofs
[edit] Necessary Condition
[edit] Sufficient Condition
[edit] Equivalence Proofs
Checking in turn each of the criteria for equivalence:
[edit] Reflexive
[edit] Symmetric
[edit] Transitive
[edit] Ordering Proofs
Checking in turn each of the criteria for an ordering:
[edit] Reflexivity
[edit] Transitivity
[edit] Antisymmetry
[edit] Group Proofs
Taking the group axioms in turn:
[edit] G0: Closure
[edit] G1: Associativity
[edit] G2: Identity
[edit] G3: Inverses
[edit] Ring Proofs
Taking the ring axioms in turn:
[edit] A: Addition forms a Group
[edit] M0: Closure of Ring Product
[edit] M1: Associativity of Ring Product
[edit] D: Distributivity of Ring Product over Addition
[edit] Proof by Mathematical Induction
Proof by induction:
For all
, let
be the proposition:
.
is true, as this just says
.
[edit] Basis for the Induction
is the case
, which has been proved above. This is our basis for the induction.
[edit] Induction Hypothesis
Now we need to show that, if
is true, where
, then it logically follows that
is true.
So this is our induction hypothesis:
.
Then we need to show:
.
[edit] Induction Step
This is our induction step:
| ||||||
|
|
So
and the result follows by the Principle of Mathematical Induction.
Therefore:
[edit] Tableau proofs
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| <line number> | <line numbers> |
| link to ProofWiki entry | <line numbers> | ||
| <line number> | <line numbers> |
| link to ProofWiki entry | <line numbers> |
...etc.
[edit] Logical Axiom references
These are for tableau proofs:
- Declaration of a Proposition: P
- Rule of Assumption: A
- Law of the Excluded Middle: LEM
[edit] Barnstars
The tireless contributor barnstar for all the long hours you have spent adding to the site. Thank you and congratulations!















