User:Prime.mover/Constructs
Useful constructs
Useful constructs for anyone to cut and paste:
Blackboard characters: $\N \Z \Q \R \C \P \S$
$fred := bert$
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 $T = \left({S, \tau}\right)$ 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 $\displaystyle \sum_{n=0}^\infty a_n \left({x - \xi}\right)^n$ 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 $\displaystyle \sum_{n=1}^\infty a_n$ be a convergent series in $\R$.
Let $\left \langle {s_n} \right \rangle$ be the sequence of partial sums of $\displaystyle \sum_{n=1}^\infty a_n$.
Let $\left \langle {x_n} \right \rangle$ be a sequence in $\R$.
Let $\left \langle {x_n} \right \rangle$ be a Cauchy sequence.
Let $\displaystyle \lim_{n \to \infty} x_n = l$.
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.
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
| 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.
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.
