User:Matt Westwood

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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

You might also catch me asking (but usually answering) questions on http://www.mathhelpforum.com/math-help/ 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.

Category:Matt Westwood's Stubs

... work in progress ...

Keep chipping away ...

{{POTW Candidate}}

mediawiki:sidebar

www.proofwiki.org/webmail

Matt Westwood (talk) (contributions) (NUM)

[edit] URM Programs

Line		Command		Comment
$1$		$Z \left({n}\right)$
$2$		$S \left({n}\right)$
$3$		$C \left({m, n}\right)$
$4$		$J \left({m, n, q}\right)$

...etc.

basic instruction

instruction pointer

null URM program

stage of computation

Let $P$ be a URM program.

Let $P$ be a normalized URM program.

Let $l = \lambda \left({P}\right)$ be the number of basic instructions in $P$ .

Let $u = \rho \left({Q}\right)$ be the number of registers used by $Q$ .

Trace Table:

Stage	Instruction	$R 1$	$R 2$	$R 3$
$0$	$1$	$r 1$	$r 2$	$r 3$
$1$	$2$	$r 1$	$r 2$	$r 3$

...etc.

[edit] Useful constructs

Useful constructs for anyone to cut and paste:

Blackboard characters: $\N \Z \Q \R \C \O \P \S$

Redirect User:Matt Westwood#Barnstars

For example:

$fred \ \stackrel {\mathbf {def}} {=\!=} \ bert$

Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space.

Let $\left \lfloor{x}\right \rfloor$ be the floor of $x$ .

Let $\left \lceil{x}\right \rceil$ be the ceiling of $x$ .

Let $T = \left({A, \vartheta}\right)$ be a topological space.

Let $M = \left({A, d}\right)$ be a metric space.

Let $N_\epsilon \left({x}\right)$ be an $ε$ -neighborhood in $M = \left({A, d}\right)$ .

Let $\xi \in \R$ be a real number.

Let $\sum_{n=0}^\infty a_n \left({x - \xi}\right)^n$ be a power series about $ξ$ .

Let $f$ be a real function which is continuous on the closed interval $\left[{a \, . \, . \, b}\right]$ and differentiable on the open interval $\left({a \, . \, . \, b}\right)$ .

Let $f$ have a primitive $F$ on $\left[{a \, . \, . \, b}\right]$ .

Let $\sum_{n=1}^\infty a_n$ be a convergent series in $\R$ .

Let $\left \langle {s_n} \right \rangle$ be the sequence of partial sums of $\sum_{n=1}^\infty a_n$ .

Let $\left \langle {x_n} \right \rangle$ be a sequence in $\R$ .

Let $\left \langle {x_n} \right \rangle$ be a Cauchy sequence.

Let $\lim_{n \to \infty} x_n = l$ .

Let $x_n \to l$ as $n \to \infty$ .

Let $\left \langle {x_{n_r}} \right \rangle$ be a subsequence of $\left \langle {x_n} \right \rangle$ .

Let $\mathbf{A} = \left[{a}\right]_{m n}$ be an $m \times n$ matrix.

Let $\mathbf{A} = \left[{a}\right]_{n}$ be a square matrix of order $n$ .

Let $\det \left({\mathbf{A}}\right)$ be the determinant of $\mathbf{A}$ .

Let $\mathcal {M}_{S} \left({m, n}\right)$ be the $m \times n$ matrix space over $S$ .

Let $\left\{{x, y, z}\right\}$ be a set.

Let $\mathcal{P} \left({S}\right)$ be the power set of the set $S$ .

Let $\left({S, \circ}\right)$ be an algebraic structure or a semigroup.

Let $\left({G, \circ}\right)$ be a group whose identity is $e$ .

Let $\left({R, +, \circ}\right)$ be a ring whose zero is $0 R$ .

Let $\left({R, +, \circ}\right)$ be a ring with unity whose zero is $0 R$ and whose unity is $1 R$ .

Let $\left({K, +, \circ}\right)$ be a division ring whose zero is $0 K$ and whose unity is $1 K$ .

Let $\left \langle {S} \right \rangle$ be the group generated by $S$ .

Let $\left \langle {g} \right \rangle = \left({G, \circ}\right)$ be a cyclic group.

Let $\left({G, +_G: \circ}\right)_R$ be an $R$ -module.

Let $\left({G, +_G: \circ}\right)_K$ be a $K$ -vector space.

Let $\left({G, +_G: \circ}\right)_R$ be a unitary $R$ -module whose dimension is finite.

Let $\mathcal {L}_R \left({G, H}\right)$ be the set of all linear transformations from $G$ to $H$ .

Let $\mathcal {L}_R \left({G}\right)$ be the set of all linear operators on $G$ .

Let $\left[{u; \left \langle {b_m} \right \rangle, \left \langle {a_n} \right \rangle}\right]$ be the matrix of $u$ relative to $\left \langle {a_n} \right \rangle$ and $\left \langle {b_m} \right \rangle$ .

Let $D \left[{x}\right]$ be the set of polynomials in $x$ over $D$ .

Let $D \left[{X}\right]$ be the ring of polynomial forms in $X$ over $D$ .

Let $P \left({D}\right)$ be the ring of polynomial functions over $D$ .

Let $G *$ be the algebraic dual of $G$ .

Let $G * *$ be the algebraic dual of $G *$ .

Let $M^\circ$ be the annihilator of $M$ .

Let $\left \langle {x, t'} \right \rangle$ be as defined in Evaluation Linear Transformation.

Let $J$ be an ideal of $R$ .

Let $\left({R / J, +, \circ}\right)$ be the quotient ring defined by $J$ .

Let $\left({D, +, \circ}\right)$ be an integral domain or a principal ideal domain whose zero is $0 D$ and whose unity is $1 D$ .

Let $\left({F, +, \circ}\right)$ be a field whose zero is $0 F$ and whose unity is $1 F$ .

Let $\left({K, +, \circ}\right)$ be a quotient field of an integral domain $\left({D, +, \circ}\right)$ .

Let $\left({D, +, \circ; \le}\right)$ be a totally ordered integral domain whose zero is $0 D$ and whose unity is $1 D$ .

Let $\left({S; \preceq}\right)$ be a totally ordered set.

Let $\left({S, \circ; \preceq}\right)$ be an ordered structure.

Let $\left({S, \circ; \preceq}\right)$ be a naturally ordered semigroup.

Let $\left({S, \circ, \ast; \preceq}\right)$ be a Naturally Ordered Semigroup with Product.

$\left[{m \, . \, . \, n}\right]$ is the closed interval between $m$ and $n$ .

$\N$ , $\N^*$ , $\N_k$ , $\N^*_k$

$\Z$ , $\Z^*$ , $\Z_+$ , $\Z^*_+$ ,

$\Q$

$\R$

$\C$

Let $\Z_m$ be the set of integers modulo $m$ .

Let $\Z'_m$ be the set of integers coprime to $m$ in $\Z_m$ .

Let $\left({\Z, +}\right)$ be the Additive Group of Integers.

Let $\left({\Z, +, \times}\right)$ be the integral domain of integers.

Let $\left({\Z_m, +_m, \times_m}\right)$ ‎ be the ring of integers modulo $m$ .

Let $\left({\Z_m, +_m}\right)$ be the Additive Group of Integers Modulo $m$ .

Let $n \Z$ be the set of integer multiples of $n$ .

Let $\left({x}\right)$ be the principal ideal of $\left({\Z, +, \times}\right)$ generated by $x$ .

Let $\operatorname{Char} \left({R}\right)$ be the characteristic of $R$ .

The cardinality of a set $S$ is written $\left|{S}\right|$ .

Let $\left \langle {s_k} \right \rangle_{k \in A}$ be a sequence in $S$ .

Let $\gcd \left\{{a, b}\right\}$ be the greatest common divisor of $a$ and $b$ .

Let $\operatorname{lcm} \left\{{a, b}\right\}$ be the lowest common multiple of $a$ and $b$ .

Let $\left|{a}\right|$ be the absolute value of $a$ .

$a \equiv b \left({\bmod\, m}\right)$ : " $a$ is congruent to $b$ modulo $m$ ."

$\left[\!\left[{a}\right]\!\right]_m$ is the residue class of $a$ (modulo $m$ ).

Let $\left[{G : H}\right]$ be the index of $H$ in $G$ .

Let $C_G \left({H}\right)$ be the centralizer of $H$ in $G$ .

Let $N_G \left({S}\right)$ be the normalizer of $S$ in $G$ .

Let $G / N$ be the quotient group of $G$ by $N$ .

Let $Z \left({G}\right)$ be the center of $G$ .

Let $x \in G$ .

Let $N_G \left({x}\right)$ be the normalizer of $x$ in $G$ .

Let $\left[{G : N_G \left({x}\right)}\right]$ be the index of $N_G \left({x}\right)$ in $G$ .

Let $S n$ denote the set of permutations on $n$ letters.

Let $S n$ denote the symmetric group on $n$ letters.

Let $\operatorname{Fix} \left({\pi}\right)$ be the set of elements fixed by $π$ .

Matrix (square brackets): $\begin{bmatrix} x & y \\ z & v \end{bmatrix}$

Matrix (round brackets): $\begin{pmatrix} x & y \\ z & v \end{pmatrix}$

two-row notation: $\begin{bmatrix} 1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 3 \end{bmatrix}$

cycle notation: $\begin{bmatrix} x & y \end{bmatrix}$

Let $\operatorname{Orb} \left({x}\right)$ be the orbit of $x$ .

Let $\operatorname{Stab} \left({x}\right)$ be the stabilizer of $x$ by $G$ .

Let $\left({S, \ast_1, \ast_2, \ldots, \ast_n: \circ}\right)_R$ be an $R$ -algebraic structure.

[edit] Ordinary proofs

		$=$
$\implies$		$=$

...etc.

[edit] Iff Proofs

[edit] Sufficient Condition

[edit] Necessary 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 $n \in \N^*$ , let $P \left({n}\right)$ be the proposition:

p r o p o s i t i o n n

.

$P (1)$ is true, as this just says $p r o p o s i t i o n 1$ .

[edit] Basis for the Induction

$P (2)$ is the case $p r o p o s i t i o n 2$ , which has been proved above. This is our basis for the induction.

[edit] Induction Hypothesis

Now we need to show that, if $P \left({k}\right)$ is true, where $k \ge 2$ , then it logically follows that $P \left({k+1}\right)$ is true.

So this is our induction hypothesis:

p r o p o s i t i o n k

.

Then we need to show:

p r o p o s i t i o n k + 1

.

[edit] Induction Step

This is our induction step:

		$=$
$\implies$		$=$

So $P \left({k}\right) \implies P \left({k+1}\right)$ and the result follows by the Principle of Mathematical Induction.

Therefore:

p r o p o s i t i o n n

[edit] Tableau proofs

Line		Pool	Formula	Rule	Depends upon	Notes
<line number>		<line numbers>	$\ldots{}$	link to ProofWiki entry	<line numbers>
<line number>		<line numbers>	$\ldots{}$	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

Rule of Conjunction: $\and \mathcal{I}$

Rule of Simplification: $\and \mathcal{E}_1$ or $\and \mathcal{E}_2$

Rule of Addition: $\or \mathcal{I}_1$ or $\or \mathcal{I}_2$

Rule of Or-Elimination: $\or \mathcal{E}$

Modus Ponendo Ponens: $\implies \mathcal{E}$

Rule of Implication: $\implies \mathcal{I}$

Rule of Not-Elimination: $\neg \mathcal{E}$

Rule of Proof by Contradiction: $\neg \mathcal{I}$

Rule of Bottom-Elimination: $\bot \mathcal{E}$

Law of the Excluded Middle: LEM

Double Negation Introduction: $\neg \neg \mathcal{I}$

Double Negation Elimination: $\neg \neg \mathcal{E}$

[edit] Barnstars

File:Tireless Contributor Barnstar.gif

The tireless contributor barnstar for all the long hours you have spent adding to the site. Thank you and congratulations!

Retrieved from "http://www.proofwiki.org/wiki/User:Matt_Westwood"

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