User:Prime.mover



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

I'm now about to tackle the foundations of complex analysis.

But first I'm going to establish some results for metric spaces.

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

Category:Matt Westwood's Stubs ... work in progress ...

Keep chipping away ...

Useful constructs for anyone to cut and paste.

Wiki LaTeX

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 critera for equivalence:

Ordering Proofs
Checking in turn each of the critera 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
This is our induction step:

$$ $$

So $$P \left({k}\right) \Longrightarrow P \left({k+1}\right)$$ 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: $\land \mathcal{I}$


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


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


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


 * Modus Ponendo Ponens: $\Longrightarrow \mathcal{E}$


 * Rule of Implication: $\Longrightarrow \mathcal{I}$


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


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


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


 * Law of the Excluded Middle: $\textrm{LEM}$


 * Double Negation Introduction: $\lnot \lnot \mathcal{I}$


 * Double Negation Elimination: $\lnot \lnot \mathcal{E}$

Barnstars


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