User:Prime.mover/Constructs

Useful constructs
Useful constructs for anyone to cut and paste:

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


 * 1) Redirect User:Prime.mover

For example:

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

Let $A$ be an algebra over the field $\R$ whose bilinear map $m: A^2 \to A$ is called multiplication

Let the unity of $A$ be $1$ such that $\forall a \in A: m \left({1, a}\right) = a = m \left({a, 1}\right)$.

We can abbreviate $m \left({a, b}\right)$ as $a b$.

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

Let $X$ be a discrete random variable on $\left({\Omega, \Sigma, \Pr}\right)$.

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 $\epsilon$-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 $\left({S, \circ, *}\right)$ be a boolean ring whose identity for $\circ$ is $e^\circ$ and whose identity for $*$ 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 $n$.

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

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

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 $\pi$.

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:
 * $proposition_n$

$P \left({1}\right)$ is true, as this just says $proposition_1$.

Basis for the Induction
$P \left({2}\right)$ 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 $P \left({k}\right) \implies P \left({k+1}\right)$ 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: $\land \mathcal I$


 * 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: $\implies \mathcal E$


 * 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: $\neg \neg \mathcal E$

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.