User:Prime.mover


 * MattWestwoodAvatar.jpg

I'm on the local email as prime.mover@proofwiki.org so drop us a line when you want.

You might also catch me asking (but usually answering) questions on http://www.mathhelpforum.com/math-help/ which I haunt when ProofWiki's down.

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

What's THIS For ...!
Sandbox, In Progress, Later, Help Needed.

Category:Matt Westwood's Stubs

... work in progress ...

help!

Keep chipping away ...

Jokes

 

Potw

mediawiki:sidebar

Wiki LaTeX

www.proofwiki.org/webmail

URM Programs
...etc.

input

output

register

terminate

basic instruction

instruction pointer

null URM program

exit jump

exit line

stage of computation

state

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:

...etc.

Useful constructs
Useful constructs for anyone to cut and paste:

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


 * 1) Redirect User:Matt Westwood

For example:

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

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 $\epsilon$-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 $$\xi$$.

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 congruence 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 $\pi$.

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.

Ordinary proofs
$$ $$

...etc.

Equivalence Proofs
Checking in turn each of the criteria for equivalence:

Ordering Proofs
Checking in turn each of the criteria for an ordering:

Group Proofs
Taking the group axioms in turn:

Ring Proofs
Taking the ring axioms in turn:

Proof by Mathematical Induction
Proof by induction:

For all $$n \in \N^*$$, let $$P \left({n}\right)$$ be the proposition $$proposition_n$$.


 * $$P(1)$$ is true, as this just says $$proposition_1$$.

Basis for the Induction

 * $$P(2)$$ is the case $$proposition_2$$, which has been proved above. This is our basis for the induction.

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:
 * $$proposition_k$$.

Then we need to show:
 * $$proposition_{k+1}$$.

Induction Step
$$ $$

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

Therefore $$proposition_n$$.

Tableau proofs
...etc.

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: $\neg \neg \mathcal{E}$

Barnstars


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