$
% macros
% MathJax
\newcommand{\bigtimes}{\mathop{\vcenter{\hbox{$\Huge\times\normalsize$}}}} % prefix with \! and postfix with \!\!
% sized grouping symbols
\renewcommand {\aa} [1] {\left\langle {#1} \right\rangle} % <> angle brackets
\newcommand {\bb} [1] {\left[ {#1} \right]} % [] brackets
\newcommand {\cc} [1] {\left\{ {#1} \right\}} % {} curly braces
\newcommand {\mm} [1] {\lVert {#1} \rVert} % || || double norm bars
\newcommand {\nn} [1] {\lvert {#1} \rvert} % || norm bars
\newcommand {\pp} [1] {\left( {#1} \right)} % () parentheses
% unit
\newcommand {\unit} [1] {\bb{\mathrm{#1}}} % measurement unit
% math
\newcommand {\fn} [1] {\mathrm{#1}} % function name
% sets
\newcommand {\setZ} {\mathbb{Z}}
\newcommand {\setQ} {\mathbb{Q}}
\newcommand {\setR} {\mathbb{R}}
\newcommand {\setC} {\mathbb{C}}
% arithmetic
\newcommand {\q} [2] {\frac{#1}{#2}} % quotient, because fuck \frac
% trig
\newcommand {\acos} {\mathrm{acos}} % \mathrm{???}^{-1} is misleading
\newcommand {\asin} {\mathrm{asin}}
\newcommand {\atan} {\mathrm{atan}}
\newcommand {\atantwo} {\mathrm{atan2}} % at angle = atan2(y, x)
\newcommand {\asec} {\mathrm{asec}}
\newcommand {\acsc} {\mathrm{acsc}}
\newcommand {\acot} {\mathrm{acot}}
% complex numbers
\newcommand {\z} [1] {\tilde{#1}}
\newcommand {\conj} [1] {{#1}^\ast}
\renewcommand {\Re} {\mathfrak{Re}} % real part
\renewcommand {\Im} {\mathrm{I}\mathfrak{m}} % imaginary part
% quaternions
\newcommand {\quat} [1] {\tilde{\mathbf{#1}}} % quaternion symbol
\newcommand {\uquat} [1] {\check{\mathbf{#1}}} % versor symbol
\newcommand {\gquat} [1] {\tilde{\boldsymbol{#1}}} % greek quaternion symbol
\newcommand {\guquat}[1] {\check{\boldsymbol{#1}}} % greek versor symbol
% vectors
\renewcommand {\vec} [1] {\mathbf{#1}} % vector symbol
\newcommand {\uvec} [1] {\hat{\mathbf{#1}}} % unit vector symbol
\newcommand {\gvec} [1] {\boldsymbol{#1}} % greek vector symbol
\newcommand {\guvec} [1] {\hat{\boldsymbol{#1}}} % greek unit vector symbol
% special math vectors
\renewcommand {\r} {\vec{r}} % r vector [m]
\newcommand {\R} {\vec{R}} % R = r - r' difference vector [m]
\newcommand {\ur} {\uvec{r}} % r unit vector [#]
\newcommand {\uR} {\uvec{R}} % R unit vector [#]
\newcommand {\ux} {\uvec{x}} % x unit vector [#]
\newcommand {\uy} {\uvec{y}} % y unit vector [#]
\newcommand {\uz} {\uvec{z}} % z unit vector [#]
\newcommand {\urho} {\guvec{\rho}} % rho unit vector [#]
\newcommand {\utheta} {\guvec{\theta}} % theta unit vector [#]
\newcommand {\uphi} {\guvec{\phi}} % phi unit vector [#]
\newcommand {\un} {\uvec{n}} % unit normal vector [#]
% vector operations
\newcommand {\inner} [2] {\left\langle {#1} , {#2} \right\rangle} % <,>
\newcommand {\outer} [2] {{#1} \otimes {#2}}
\newcommand {\norm} [1] {\mm{#1}}
\renewcommand {\dot} {\cdot} % dot product
\newcommand {\cross} {\times} % cross product
% matrices
\newcommand {\mat} [1] {\mathbf{#1}} % matrix symbol
\newcommand {\gmat} [1] {\boldsymbol{#1}} % greek matrix symbol
% ordinary derivatives
\newcommand {\od} [2] {\q{d #1}{d #2}} % ordinary derivative
\newcommand {\odn} [3] {\q{d^{#3}{#1}}{d{#2}^{#3}}} % nth order od
\newcommand {\odt} [1] {\q{d{#1}}{dt}} % time od
% partial derivatives
\newcommand {\de} {\partial} % partial symbol
\newcommand {\pd} [2] {\q{\de{#1}}{\de{#2}}} % partial derivative
\newcommand {\pdn} [3] {\q{\de^{#3}{#1}}{\de{#2}^{#3}}} % nth order pd
\newcommand {\pdd} [3] {\q{\de^2{#1}}{\de{#2}\de{#3}}} % 2nd order mixed pd
\newcommand {\pdt} [1] {\q{\de{#1}}{\de{t}}} % time pd
% vector derivatives
\newcommand {\del} {\nabla} % del
\newcommand {\grad} {\del} % gradient
\renewcommand {\div} {\del\dot} % divergence
\newcommand {\curl} {\del\cross} % curl
% differential vectors
\newcommand {\dL} {d\vec{L}} % differential vector length [m]
\newcommand {\dS} {d\vec{S}} % differential vector surface area [m^2]
% special functions
\newcommand {\Hn} [2] {H^{(#1)}_{#2}} % nth order Hankel function
\newcommand {\hn} [2] {H^{(#1)}_{#2}} % nth order spherical Hankel function
% transforms
\newcommand {\FT} {\mathcal{F}} % fourier transform
\newcommand {\IFT} {\FT^{-1}} % inverse fourier transform
% signal processing
\newcommand {\conv} [2] {{#1}\ast{#2}} % convolution
\newcommand {\corr} [2] {{#1}\star{#2}} % correlation
% abstract algebra
\newcommand {\lie} [1] {\mathfrak{#1}} % lie algebra
% other
\renewcommand {\d} {\delta}
% optimization
%\DeclareMathOperator* {\argmin} {arg\,min}
%\DeclareMathOperator* {\argmax} {arg\,max}
\newcommand {\argmin} {\fn{arg\,min}}
\newcommand {\argmax} {\fn{arg\,max}}
% waves
\renewcommand {\l} {\lambda} % wavelength [m]
\renewcommand {\k} {\vec{k}} % wavevector [rad/m]
\newcommand {\uk} {\uvec{k}} % unit wavevector [#]
\newcommand {\w} {\omega} % angular frequency [rad/s]
\renewcommand {\TH} {e^{j \w t}} % engineering time-harmonic function [#]
% classical mechanics
\newcommand {\F} {\vec{F}} % force [N]
\newcommand {\p} {\vec{p}} % momentum [kg m/s]
% \r % position [m], aliased
\renewcommand {\v} {\vec{v}} % velocity vector [m/s]
\renewcommand {\a} {\vec{a}} % acceleration [m/s^2]
\newcommand {\vGamma} {\gvec{\Gamma}} % torque [N m]
\renewcommand {\L} {\vec{L}} % angular momentum [kg m^2 / s]
\newcommand {\mI} {\mat{I}} % moment of inertia tensor [kg m^2/rad]
\newcommand {\vw} {\gvec{\omega}} % angular velocity vector [rad/s]
\newcommand {\valpha} {\gvec{\alpha}} % angular acceleration vector [rad/s^2]
% electromagnetics
% fields
\newcommand {\E} {\vec{E}} % electric field intensity [V/m]
\renewcommand {\H} {\vec{H}} % magnetic field intensity [A/m]
\newcommand {\D} {\vec{D}} % electric flux density [C/m^2]
\newcommand {\B} {\vec{B}} % magnetic flux density [Wb/m^2]
% potentials
\newcommand {\A} {\vec{A}} % vector potential [Wb/m], [C/m]
% \F % electric vector potential [C/m], aliased
% sources
\newcommand {\I} {\vec{I}} % line current density [A] , [V]
\newcommand {\J} {\vec{J}} % surface current density [A/m] , [V/m]
\newcommand {\K} {\vec{K}} % volume current density [A/m^2], [V/m^2]
% \M % magnetic current [V/m^2], aliased, obsolete
% materials
\newcommand {\ep} {\epsilon} % permittivity [F/m]
% \mu % permeability [H/m], aliased
\renewcommand {\P} {\vec{P}} % polarization [C/m^2], [Wb/m^2]
% \p % electric dipole moment [C m], aliased
\newcommand {\M} {\vec{M}} % magnetization [A/m], [V/m]
\newcommand {\m} {\vec{m}} % magnetic dipole moment [A m^2]
% power
\renewcommand {\S} {\vec{S}} % poynting vector [W/m^2]
\newcommand {\Sa} {\aa{\vec{S}}_t} % time averaged poynting vector [W/m^2]
% quantum mechanics
\newcommand {\bra} [1] {\left\langle {#1} \right|} % <|
\newcommand {\ket} [1] {\left| {#1} \right\rangle} % |>
\newcommand {\braket} [2] {\left\langle {#1} \middle| {#2} \right\rangle}
$
This is a review of set theory. This needs a lot of work.
boolean algebra
| $A$ |
$B$ |
$F$ |
$T$ |
$\neg A$ |
$\neg B$ |
$\vee$ |
$\wedge$ |
$\rightarrow$ |
$\leftrightarrow$ |
$\leftarrow$ |
$\downarrow$ |
$\uparrow$ |
$\nrightarrow$ |
$\nleftrightarrow$ |
$\nleftarrow$ |
| F |
F |
F |
T |
T |
T |
F |
F |
T |
T |
T |
T |
T |
F |
F |
F |
| F |
T |
F |
T |
T |
F |
T |
F |
T |
F |
F |
F |
T |
F |
T |
T |
| T |
F |
F |
T |
F |
T |
T |
F |
F |
F |
T |
F |
T |
T |
T |
F |
| T |
T |
F |
T |
F |
F |
T |
T |
T |
T |
T |
F |
F |
F |
F |
F |
$\neg, \vee, \wedge$ can synthesize any other column in the truth table. brackets $[]$ can be used to unambiguously construct compound expressions. defining a precedence order relaxes the use of brackets. $\neg$ first, then $\wedge$, then $\vee$.
logical consequence
$$
\Rightarrow
$$
logical equivalence
$$
\Leftrightarrow
$$
definition
$$
\equiv
$$
set
$$
A = \cc{1, a, \bigstar, \triangle}
$$
membership
$$
1 \in A
$$
$$
2 \notin A
$$
cardinality
$$
\nn{A} =\,\text{number of elements}
$$
quantifiers
$$
\forall x \in A [\fn{Predicate}(x)]
$$
$$
\exists x \in A [\fn{Predicate}(x)]
$$
$$
\neg \forall x \in A [\fn{Predicate}(x)] \Leftrightarrow \exists x \in A [\neg \fn{Predicate}(x)]
$$
$$
\neg \exists x \in A [\fn{Predicate}(x)] \Leftrightarrow \forall x \in A [\neg \fn{Predicate}(x)]
$$
subset
$$
A \subseteq B \Leftrightarrow \forall x \in A [x \in B]
$$
set equality
$$
A = B \Leftrightarrow A \subseteq B \wedge B \subseteq A
$$
proper subset
$$
A \subset B \Leftrightarrow A \subseteq B \wedge A \neq B
$$
universal set
$$
A \subseteq U
$$
null set
$$
\varnothing \equiv \cc{}
$$
set builder
$$
A = \cc{x \in B | \mathrm{Predicate(x)}}
$$
set union
$$
A \cup B \equiv \cc{x \in U | x \in A \vee x \in B}
$$
set intersection
$$
A \cap B \equiv \cc{x \in U | x \in A \wedge x \in B}
$$
set subtraction
$$
A - B \equiv \cc{x \in U | a \in A \vee a \notin B}
$$
partition
ordered pair (tuple)
$$
(a, b) \equiv \cc{\cc{a}, \cc{a, b}}
$$
Cartesian product
$$
A \times B \equiv \cc{(a, b) | a \in A \wedge b \in B}
$$
$$
A^n \equiv \!\bigtimes_{i=1}^n\!\! A = \underbrace{A \cross A \cross \dots A}_{n \text{ times}}
$$
relation
$$
a R b \in R \subseteq A \times B
$$
equivalence relation
$$
\begin{matrix}
&a \sim a &\text{(reflexive)} \\
&a \sim b \Rightarrow b \sim a &\text{(symmetric)} \\
&a \sim b \wedge b \sim c \Rightarrow a \sim c &\text{(transitive)} \\
\end{matrix}
$$
total order
$$
\begin{matrix}
&a \leq b \vee b \leq a &\text{(connex)} \\
&a \leq b \wedge b \leq a \Rightarrow a = b &\text{(antisymmetric)} \\
&a \leq b \wedge b \leq c \Rightarrow a \leq c &\text{(transitive)} \\
\end{matrix}
$$
$$
a < b \equiv a \leq b \wedge a \neq b
$$
$$
a \geq b \equiv b \leq a
$$
$$
a > b \equiv b \leq a \wedge a \neq b
$$
interval
$$
[a,b] \subset A \equiv \cc{x \in A | a \leq x \leq b}
$$
function
$$
\forall a \in f [a f b \wedge a f c \Rightarrow b = c]
$$
$$
f(a) = b
$$
function mapping from domain A to range B
$$
f: A \rightarrow B
$$
unary operation
$$
f: A \rightarrow A
$$
binary operation
$$
f: A \times A \rightarrow A
$$