---
description: Formal methods reference
---

# latex

### Variable Names

| Concept                   | Math                   | LaTeX                    | Notes                                             |
| ------------------------- | ---------------------- | ------------------------ | ------------------------------------------------- |
| Sets                      | A,B,C                  | `A,B,C`                  | upper-case letters                                |
| Indexed sets              | A1, A2, …, An          | `A_1, A_2, \ldots, A_n`  | the index may be of any type, not nec. an integer |
| Statements                | P, Q, R, S             | `P, Q, R, S`             |                                                   |
| Statements with variables | P(x), Q(x), R(x), S(x) | `P(x), Q(x), R(x), S(x)` |                                                   |

### Set Theory

| Concept                                 | Math                                  | LaTeX                                                                             | Notes                                                                                 |
| --------------------------------------- | ------------------------------------- | --------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------- |
| Set                                     | {a,b,c}                               | `\{a,b,c\}`                                                                       | set enumeration                                                                       |
| Set                                     | {x∣P(x)} or {x:P(X)}                  | `\{\,x \mid P(x)\,\}` or `\{\,x : P(X)\,\}`                                       | set comprehension (described implicitly with the property P) aka set-builder notation |
| Infinite set                            | {1, 2, 3, …}                          | `\{1,2,3,\ldots\}`                                                                | and so on                                                                             |
| Infinite set                            | {…, 1, 2, 3, …}                       | `\{\ldots,1,2,3,\ldots\}`                                                         | and so on                                                                             |
| Empty set                               | ∅ or {}                               | `\varnothing` or `\emptyset` or `\{\}`                                            |                                                                                       |
| Powerset                                | P(S)                                  | `\mathbb{P}(S)` or `\mathcal{P}(S)`                                               | set of all possible subsets of set S, \|P(S)\| = 2^\|S\|                              |
| Member                                  | x∈A                                   | `x \in A`                                                                         |                                                                                       |
| Not member                              | x∉A                                   | `x \not\in A`                                                                     |                                                                                       |
| Subset                                  | A⊂B or A⊆B                            | `A \subset B` or `A \subseteq B`                                                  |                                                                                       |
| Not subset                              | A⊄B or A⊈B                           | `A \not\subset B` or `A \nsubseteq B`                                             |                                                                                       |
| Set equality                            | A=B                                   | `A = B`                                                                           | A⊆B and B⊆A                                                                           |
| Union                                   | A∪B                                   | `A \cup B`                                                                        |                                                                                       |
| Intersection                            | A∩B                                   | `A \cap B`                                                                        |                                                                                       |
| Distributed or generalized union        | ⋃SS                                   | `\bigcup SS`                                                                      | over set of sets                                                                      |
| Distributed or generalized intersection | ⋂SS                                   | `\bigcap SS`                                                                      | over set of sets                                                                      |
| Relative complement                     | A−B or A∖B                            | `A - B` or `A \setminus B`                                                        | {x∣x∈A and x∉B}                                                                       |
| Absolute complement                     | A+B                                   | `A + B`                                                                           | (A−B)∪(B−A)(A−B)∪(B−A)                                                                |
| Disjoint sets                           |                                       |                                                                                   | A∩B = ∅                                                                               |
| Set cardinality (aka size)              | card(S) or \|S\|                      | `\mathbf{card}(S)` or `\|S\|`                                                     | number of members of S                                                                |
| Boolean set                             | B or {true,false} or {tt,ff} or {1,0} | `\mathbb{B}` or `\{\mathrm{true}, \mathrm{false}\}` or `\{tt, ff\}` or `\{1, 0\}` |                                                                                       |
| Natural numbers set                     | N or N+                               | `\mathbb{N}` or `\mathbb{N}^+`                                                    | N+ = N∖{0}                                                                            |
| Integer numbers set                     | Z                                     | `\mathbb{Z}`                                                                      |                                                                                       |
| Rational numbers set                    | Q                                     | `\mathbb{Q}`                                                                      | a/b, a - numerator, b - denominator, b≠0                                              |
| Real numbers set                        | R                                     | `\mathbb{R}`                                                                      |                                                                                       |
| Set of prime numbers                    | N                                     | `\mathbb{N}`                                                                      |                                                                                       |
| Set of irrational numbers               | II                                    | `\mathbb{I}`                                                                      |                                                                                       |
| Set of complex numbers                  | CC                                    | `\mathbb{C}`                                                                      |                                                                                       |

### Cartesian Products

| Concept           | Math      | LaTeX                          | Notes                              |
| ----------------- | --------- | ------------------------------ | ---------------------------------- |
| Ordered Pair      | (a,b)     | `(a,b)`                        |                                    |
| Triple            | (a,b,c)   | `(a,b,c)`                      |                                    |
| N-tuple           | (e1,…,en) | `(e_1, \ldots, e_n)`           |                                    |
| Cartesian product | A×B       | `A \times B`                   | {(a,b)∣a∈A,b∈B}                    |
| Cartesian plane   | R×R       | `\mathbb{R} \times \mathbb{R}` | {(x,y)∣x,y∈R}                      |
| Cartesian power   | An        | `A^n`                          | A×A×…×A={x1,x2,…,xn\|x1,x2,…,xn∈A} |

### Types

| Concept                         | Math                | LaTeX                                                               | Notes                                                     |
| ------------------------------- | ------------------- | ------------------------------------------------------------------- | --------------------------------------------------------- |
| User defined type               | DEFINEDTYPE={2,4,8} | `\mathrm{DEFINEDTYPE} = \{2,4,8\}`                                  | a set defined explicitly, or by set comprehension         |
| Product type                    | B×B                 | `\mathbb{B} \times \mathbb{B}`                                      | a set of tuples (here, {(tt,tt),(tt,ff),(ff,ff),(ff,tt)}) |
| Two-argument function signature | func:R×N→N          | `\textrm{func}:\mathbb{R} \times \mathbb{N} \rightarrow \mathbb{N}` | type of arguments is product type                         |
| Enumerated type                 | signal=r \| y \| g  | `\textrm{signal} = \textrm{r}~\|~\textrm{y}~\|~\textrm{g}`          | type union of literals                                    |

### Mathematical Operators

| Concept                     | Math        | LaTeX                        | Notes  |
| --------------------------- | ----------- | ---------------------------- | ------ |
| Absolute operator           | abs −20=20  | `\mathbf{abs}\ {-20} = 20`   | unary  |
| Floor/integer part operator | floor 2.5=2 | `\mathbf{floor}\ 2.5 = 2`    | unary  |
| Integer division operator   | 5 div 2=2   | `5\ \mathbf{div}\ 2 = 2`     | binary |
| Remainder operator          | 5 rem −2=1  | `5\ \mathbf{rem}\ {-2} = 1`  | binary |
| Modulus operator            | 5 mod −2=−1 | `5\ \mathbf{mod}\ {-2} = -1` | binary |
| Equality                    | a=b         | `a = b`                      | binary |
| Inequality                  | a≠b         | `a \neq b`                   | binary |
| Less than                   | a>b         | `a > b`                      | binary |
| Less than or equal          | a≥b         | `a \ge b`                    | binary |
| More than                   | a\<b        | `a < b`                      | binary |
| More than or equal          | a≤b         | `a \le b`                    | binary |

### Bags

| Concept          | Math                                                      | LaTeX                                                                          | Notes                                                                                                 |
| ---------------- | --------------------------------------------------------- | ------------------------------------------------------------------------------ | ----------------------------------------------------------------------------------------------------- |
| Bag              | \[\[a,b,b,c,d]]                                           | `[\![ a,b,b,c,d ]\!]`                                                          | Members can repeat, unordered, also `\llbracket` and `\rrbracket` from package `stmaryrd` look better |
| Count operator   | count\[\[a,b,b,c,d]]b=2                                   | `\mathbf{count}[\![ a,b,b,c,d ]\!] b = 2`                                      | How many times element bb occurs in bag                                                               |
| Bag cardinality  | card(B)                                                   | `\mathbf{card}(B)`                                                             | Number of members of B                                                                                |
| Bag union (1)    | \[\[a,b,b,c,d]]∪\[\[a,b,c,c,d]]=\[\[a,b,b,c,c,d]]         | `[\![ a,b,b,c,d ]\!] \cup [\![ a,b,c,c,d ]\!] = [\![ a,b,b,c,c,d ]\!]`         | Maximum number an element occurs                                                                      |
| Bag union (2)    | \[\[a,b,b,c,d]]∪\[\[a,b,c,c,d]]=\[\[a,a,b,b,b,c,c,c,d,c]] | `[\![ a,b,b,c,d ]\!] \cup [\![ a,b,c,c,d ]\!] = [\![ a,a,b,b,b,c,c,c,d,c ]\!]` | Sum of elements                                                                                       |
| Bag intersection | \[\[a,b,b,c,d]]∩\[\[a,b,c,c,d]]=\[\[a,b,c,d]]             | `[\![ a,b,b,c,d ]\!] \cap [\![ a,b,c,c,d ]\!] = [\![ a,b,c,d ]\!]`             | Minimum number an element occurs                                                                      |
| Bag difference   | \[\[a,b,b,c,d]]−\[\[a,b,c,c,d]]=\[\[b,c]]                 | `[\![ a,b,b,c,d ]\!] - [\![ a,b,c,c,d ]\!] = [\![ b,c ]\!]`                    | Subtract the number of elements in the second bag from the first                                      |

### Sequences

| Concept                            | Math                       | LaTeX                           | Notes                                                            |
| ---------------------------------- | -------------------------- | ------------------------------- | ---------------------------------------------------------------- |
| Empty sequence                     | \[]                        | \[]                             |                                                                  |
| Sequence (explicit declaration)    | \[a,b,b,d]                 | `[ a,b,b,d ]`                   | In that order. This is the typical representation.               |
| Sequence (implicit declaration)    | \[e∣P(e)]                  | `[\,e \mid P(e)\,]`             | Set comprehension (described implicitly with the property P)     |
| Head of a sequence                 | hd A                       | `\mathbf{hd}~A`                 | hd \[a,b,b,c]=a                                                  |
| Tail of a sequence                 | tl A                       | `\mathbf{tl}~A`                 | tl \[a,b,b,c]=\[b,b,c]                                           |
| Length of a sequence               | len A                      | `\mathbf{len}~A`                | len \[a,b,b,c]=4                                                 |
| Element selection (by application) | A(n)                       | `A(n)`                          | Select the nn-th element of the sequence                         |
| Indices of a sequence              | inds A                     | `\mathbf{inds}~A`               | inds \[a,b,c]={1,2,3}                                            |
| Elements of a sequence             | elems A                    | `\mathbf{elems}~A`              | elems \[a,b,b,c]={a,b,c}                                         |
| Sequence concatenation             |                            |                                 | Symbol varies between notations                                  |
| List                               | \<a,b,c> or (a,b,c) or abc | `<a,b,c>` or `(a,b,c)` or `abc` | An alternative notation and naming used by some formal notations |
| Empty list                         | ()                         | `()`                            |                                                                  |

### Formal Logic

| Concept           | Math       | LaTeX                                              | Notes                                                      |
| ----------------- | ---------- | -------------------------------------------------- | ---------------------------------------------------------- |
| Conjunction (AND) | ∧or & or . | `\wedge` or `\&` or `.`                            | Alternatively `\with` from package `cmll` instead of `\&`. |
| Alternative (OR)  | ∨or \|or + | `\vee` or `\|` or `+`                              |                                                            |
| Negation (NOT)    | ¬ or ∼     | `\neg` or `\sim`                                   |                                                            |
| Implication       | →or ⇒ or ⊃ | `\rightarrow` or `\Rightarrow` or `\supset`        | premise→conclussion                                        |
| Equivalence       | ↔or ⇔ or ≡ | `\leftrightarrow` or `\Leftrightarrow` or `\equiv` |                                                            |

#### Equivalence Laws

| Concept                                   | Math                  | LaTeX                                                                     | Notes |
| ----------------------------------------- | --------------------- | ------------------------------------------------------------------------- | ----- |
| Commutation (conjunctions)                | (P∧Q) ≡ (Q∧P)         | `(P \wedge Q) \equiv (Q \wedge P)`                                        |       |
| Commutation (alternatives)                | (P∨Q) ≡ (Q∨P)         | `(P \vee Q) \equiv (Q \vee P)`                                            |       |
| Commutation (equivalences)                | (P↔Q) ≡ (Q↔P)         | `(P \leftrightarrow Q) \equiv (Q \leftrightarrow P)`                      |       |
| Association (conjunctions)                | P∧(Q∧R) ≡ (P∧Q)∧R     | `P \wedge (Q \wedge R) \equiv (P \wedge Q) \wedge R`                      |       |
| Association (alternatives)                | P∨(Q∨R) ≡ (P∨Q)∨R     | `P \vee (Q \vee R) \equiv (P \vee Q) \vee R`                              |       |
| Distribution (1)                          | P∨(Q∧R) ≡ (P∨Q)∧(P∨R) | `P \vee (Q \wedge R) \equiv (P \vee Q) \wedge (P \vee R)`                 |       |
| Distribution (2)                          | P∧(Q∨R) ≡ (P∧Q)∨(P∧R) | `P \wedge (Q \vee R) \equiv (P \wedge Q) \vee (P \wedge R)`               |       |
| De Morgan’s Law (negation of conjunction) | ¬(P∧Q) ≡ ¬P∨¬Q        | `\neg(P \wedge Q) \equiv \neg P \vee \neg Q`                              |       |
| De Morgan’s Law (negation of alternative) | ¬(P∨Q) ≡ ¬P∧¬Q        | `\neg(P \vee Q) \equiv \neg P \wedge \neg Q`                              |       |
| Double Negation                           | ¬¬P ≡ P               | `\neg\neg P \equiv P`                                                     |       |
| Excluded Middle                           | P∨¬P ≡ tt             | `P \vee \neg P \equiv tt`                                                 |       |
| Contradiction                             | P∧¬P ≡ ff             | `P \wedge \neg P \equiv ff`                                               |       |
| Material Implication                      | P→Q ≡ ¬P∨Q            | `P \rightarrow Q \equiv \neg P \vee Q`                                    |       |
| Material Equivalence (1)                  | (P↔Q) ≡ (P→Q)∧(Q→P)   | `(P \leftrightarrow Q) \equiv (P \rightarrow Q) \wedge (Q \rightarrow P)` |       |
| Material Equivalence (2)                  | (P↔Q) ≡ (P∧Q)∨(¬Q∧¬P) | `(P \leftrightarrow Q) \equiv (P \wedge Q) \vee (\neg Q \wedge \neg P)`   |       |
| Exportation                               | (P∧Q)→R ≡ P→(Q→R)     | `(P \wedge Q) \rightarrow R \equiv P \rightarrow (Q \rightarrow R)`       |       |
| Or-Simplification                         | P∨P ≡ P               | `P \vee P \equiv P`                                                       |       |
| Or-Simplification (true)                  | P∨tt ≡ tt             | `P \vee tt \equiv tt`                                                     |       |
| Or-Simplification (false)                 | P∨ff ≡ P              | `P \vee ff \equiv P`                                                      |       |
| And-Simplification                        | P∧P ≡ P               | `P \wedge P \equiv P`                                                     |       |
| And-Simplification (true)                 | P∧tt ≡ P              | `P \wedge tt \equiv P`                                                    |       |
| And-Simplification (false)                | P∧ff ≡ ff             | `P \wedge ff \equiv ff`                                                   |       |

#### Logic Forms

| Concept                | Math                   | LaTeX                                                                                  | Notes                                          |
| ---------------------- | ---------------------- | -------------------------------------------------------------------------------------- | ---------------------------------------------- |
| Modus Ponens           | P→Q, P ⊢ Q             | `P \rightarrow Q, P \vdash Q`                                                          | Andrew Harry uses `\uptherefore` from MnSymbol |
| Modus Tollens          | P→Q, ¬Q ⊢ ¬P           | `P \rightarrow Q, \neg Q \vdash \neg P`                                                |                                                |
| Hypothetical Syllogism | P→Q, Q→R ⊢ P→R         | `P \rightarrow Q, Q \rightarrow R \vdash P \rightarrow R`                              |                                                |
| Constructive Dillema   | P→Q∧R→S, P∨R ⊢ Q∨S     | `P \rightarrow Q \wedge R \rightarrow S, P \vee R \vdash Q \vee S`                     |                                                |
| Destructive Dillema    | P→Q∧R→S, ¬Q∨¬S ⊢ ¬P∨¬R | `P \rightarrow Q \wedge R \rightarrow S, \neg Q \vee \neg S \vdash \neg P \vee \neg R` |                                                |
| Conjunction            | P, Q ⊢ P∧Q             | `P, Q \vdash P \wedge Q`                                                               |                                                |
| Addition               | P ⊢ P∨Q                | `P \vdash P \vee Q`                                                                    |                                                |

#### Predicate Logic

| Concept                           | Math            | LaTeX                                                         | Notes                                                        |
| --------------------------------- | --------------- | ------------------------------------------------------------- | ------------------------------------------------------------ |
| Universal qualifier (all)         | ∀x              | `\forall_x`                                                   |                                                              |
| Universal specification           | ∀x∙Px           | `\forall_x \bullet P_x`                                       | ∀x∙man(x)→mortal(x)                                          |
| Existential qualifier (exists)    | ∃x              | `\exists_x`                                                   |                                                              |
| Existential specification         | ∃x∙Px           | `\exists_x \bullet P_x`                                       | ∃x∙Socrates(x)∧man(x)∃x∙Socrates(x)∧man(x)                   |
| Ranging over types (1)            | ∀x∈X or ∀x:X    | `\forall_x \in X` or `\forall_x : X`                          | The former suggests X is a set, the latter that it is a type |
| Ranging over types (2)            | ∃x∈X or ∃x:X    | `\exists_x \in X` or `\exists_x : X`                          | The former suggests X is a set, the latter that it is a type |
| Covnersion between qualifiers (1) | ∀x∙Px ≡ ¬∃x∙¬Px | `\forall_x \bullet P_x \equiv \neg\exists_x \bullet \neg P_x` | Alternatively ∄x                                            |
| Covnersion between qualifiers (2) | ¬∀x∙Px ≡ ∃x∙¬Px | `\neg\forall_x \bullet P_x \equiv \exists_x \bullet \neg P_x` |                                                              |
| Covnersion between qualifiers (3) | ¬∀x∙¬Px ≡ ∃x∙Px | `\neg\forall_x \bullet \neg P_x \equiv \exists_x \bullet P_x` |                                                              |
| Covnersion between qualifiers (4) | ∀x∙¬Px ≡ ¬∃x∙Px | `\forall_x \bullet \neg P_x \equiv \neg\exists_x \bullet P_x` | Alternatively ∄x                                            |

### Relationships and Functions

| Concept                          | Math                                     | LaTeX                                                       | Notes                                                                                                                    |
| -------------------------------- | ---------------------------------------- | ----------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------ |
| A mapping                        | x↦y or (x,y)                             | `x \mapsto y` or `(x, y)`                                   |                                                                                                                          |
| Function signiature              | f: X→Y                                   | `f:X \rightarrow Y`                                         | Mapping from X (domain) to Y (co-domain). What is allowed to be in X, Y is described by the precondition, postcondition. |
| Function definition              | square(x)=x2 where 1≤x≤5                 | `\mathbf{square}(x) = x^2\ \text{where}\ 1 \le x \le 5`     |                                                                                                                          |
| Function defined as co-ordinates | square={(1,2),(2,4),(3,9),(4,16),(5,25)} | `\mathbf{square} = \{(1,2), (2,4), (3,9), (4,16), (5,25)\}` | The set of right coordinates (here {2,4,9,16,25}) is the range of a function                                             |
| Range                            | range(f)                                 | `\mathrm{range}(f)`                                         | range({(1,a),(2,b),(3,c)})={a,b,c}                                                                                       |
| Partial function signiature      | f: X↛ Y                                  | `f:X \nrightarrow Y`                                        | Not all elements from X are mapped to elements from Y                                                                    |

### Transactional Memory

| Concept                    | Math       | LaTeX               | Notes                                                                                          |
| -------------------------- | ---------- | ------------------- | ---------------------------------------------------------------------------------------------- |
| History                    | H          | `H`                 |                                                                                                |
| Sequential witness history | S or S^    | `S` or `\hat{S}`    |                                                                                                |
| Completion                 |            |                     |                                                                                                |
| T-objects                  |            |                     |                                                                                                |
| Transaction                | Ti, Tj, Tk | `T_i`, `T_j`, `T_k` |                                                                                                |
| Subhistory – transaction   | H\|Ti      | `H\|T_i`            | A subhistory containing all operations in history HH that are executed within transaction TiTi |
| Subhistory – t-object      | H\|x       | `H\|x`              | A subhistory containing all operations in history HH that are executed on t-object xx          |
| History prefix             | H=H′∘H′    | `H = H' \circ H''`  | H′H′ is a prefix of H′′H″                                                                      |
| Commit                     |            |                     |                                                                                                |
| Abort                      |            |                     |                                                                                                |
| Forced abort               |            |                     |                                                                                                |
| Read                       |            |                     |                                                                                                |
| Write                      |            |                     |                                                                                                |
| Operation                  | π          | `\pi`               |                                                                                                |

#### Notation for Presenting Histories

| Concept                                  | Math                       | LaTeX                                                           | Notes                                                                                          |
| ---------------------------------------- | -------------------------- | --------------------------------------------------------------- | ---------------------------------------------------------------------------------------------- |
| Variables                                | x, y, z                    | `x,y,z`                                                         |                                                                                                |
| Variable local copy or buffer            | x\_                        | `\underline{x}`                                                 | Transaction’s copy of x                                                                        |
| Variable versioning                      | xn                         | `\overset{n}{x}`                                                | Variable x with version n                                                                      |
| Transaction identifiers                  | Ti, Tj, Tk                 | `T_i`, `T_j`, `T_k`                                             |                                                                                                |
| Retried transaction identifiers          | T′i, T′′i, T′′′i           | `T_i'`, `T_i''`, `T_i'''`                                       | Transaction Ti‘s first, second, etc. retry                                                     |
| Read operation                           | r(x)v                      | `r(x)v`                                                         | Reads value vv from variable xx, can be used with specific values, e.g., r(x)1                 |
| Write operation                          | w(x)v                      | `w(x)v`                                                         | Writes value vv to variable xx, can be used with specific values, e.g., w(x)1                  |
| Read-write operation                     | {x→vy} or {y←vx}           | `\{x \xrightarrow{v} y\}` or `\{y \xleftarrow{v} x\}`           | Shorthand for the sequence r(x)v,w(y)v                                                         |
| Read operation with explicit membership  | ri(x)v                     | `r_i(x)v`                                                       | Read operation executed within Ti                                                              |
| Write operation with explicit membership | wi(x)v                     | `w_i(x)v`                                                       | Write operation executed within Ti                                                             |
| Operation identifier                     | op, opr, opw, opi          | `\mathit{op}, \mathit{op}^r, \mathit{op}^w, \mathit{op}_i`      | Operation, read operation, write operation, operation within Ti                                |
| Operation membership                     | op∈Ti                      | `\mathit{op} \in T_i`                                           | Shorthand for op∈H\|Ti                                                                         |
| Transaction start                        | \[\[ or \llbracket         | `[\![` or `\llbracket`                                          | \llbracket requires the stmaryrd package                                                       |
| Transaction commit                       | ]] or \rrbracket           | `]\!]` or `\rrbracket`                                          | \rrbracket requires the stmaryrd package                                                       |
| Transaction abort                        | ↻ or \righttoleftarrowor ↶ | `\circlearrowright` or `\righttoleftarrow` or `\curvearrowleft` | in amsmath or mathabx or amssymb, respectively                                                 |
| Weak transaction start                   | \[                         | `[`                                                             | Used to indicate transactions with relaxed properties, e.g. eventually consistent transactions |
| Weak transaction commit                  | ]                          | `]`                                                             | Used to indicate transactions with relaxed properties, e.g. eventually consistent transactions |
| Irrevocable operation                    | irr or ir                  | `\mathit{irr}` or `\mathit{ir}`                                 |                                                                                                |
| Happens-before relation                  | ↘                          | `\searrow`                                                      | Often rotated by about 22 degrees to be more visually appealing                                |
| Point in time                            | τ                          | `\tau`                                                          | E.g. transaction Ti starts at time τi                                                          |
| State at point in time                   | {x=1,y=2,z=3}              | `\{ x\!=\!1,y\!=\!2,z\!=\!3 \}`                                 |                                                                                                |

### Theorems

| Concept     | Math                                 | LaTeX                                 | Notes |
| ----------- | ------------------------------------ | ------------------------------------- | ----- |
| Proof       | \begin{proof} ... \end{proof}        | `\begin{proof} ... \end{proof}`       |       |
| Named proof | \begin{proof}\[name] ... \end{proof} | `\begin{proof}[name] ... \end{proof}` |       |

### Various

| Concept               | Math              | LaTeX                                | Notes                                      |
| --------------------- | ----------------- | ------------------------------------ | ------------------------------------------ |
| Binominal coefficient | (nk) = n!k!(n−k)! | `\binom{n}{k} = \frac{n!}{k!(n-k)!}` | aka choose function (pl. _symbol Newtona_) |
| Divisor               | a\|b              | `a \| b`                             | ax=b and a, b, x ∈ Z                       |

### Sources

* [Konrad Siek LaTex Formal Methods Reference](http://www.cs.put.poznan.pl/ksiek/latexmath.html)
* [OEIS Foundation LaTex mathematical symbols](https://oeis.org/wiki/List_of_LaTeX_mathematical_symbols)
* [The Comprehensive LATEX Symbol List](http://linorg.usp.br/CTAN/info/symbols/comprehensive/symbols-a4.pdf)
* [Glossary of mathematical notation and terminology](https://www.cl.cam.ac.uk/teaching/0809/CompTheory/comt09.pdf)
* Handwriting characters tool to find most suitable latex symbol: [Detexify](http://detexify.kirelabs.org/classify.html)
