# Symbols:Abbreviations

## Contents

## A

### AC

- The axiom of choice.

Same as AoC.

### ACC

### acos

- Abbreviation for
**arccosine**.

### acosec

- Abbreviation for
**arccosecant**.

### acosech

- Abbreviation for
**arc-cosech**.

### acosh

- Abbreviation for
**arc-cosh**.

### acot

- Abbreviation for
**arccotangent**.

### acoth

- Abbreviation for
**arc-coth**.

### acsc

- Abbreviation for
**arccosecant**.

### acsch

- Abbreviation for
**arc-cosech**.

### actn

- Abbreviation for
**arccotangent**.

### actnh

- Abbreviation for
**arc-coth**.

### adj

### a.e.

- Abbreviation for
**almost everywhere**.

### agm

- Abbreviation for
**arithmetic-geometric mean.**

### alg.

### aln

- Abbreviation for the
**antilogarithm**of the**natural logarithm**.

### amp

- Abbreviation for
**amplitude**.

### AoC

- The axiom of choice.

Same as AC.

### arg

- Abbreviation for
**argument (of complex number)**.

## B

### BNF

- Backus-Naur Form (previously
**Backus Normal Form**until the syntax was simplified by Peter Naur).

It was Donald Knuth who suggested the name change, on the grounds that "normal" is an inaccurate description.

### BPI

## C

### c.d.f.

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ be a random variable on $\struct {\Omega, \Sigma, \Pr}$.

The **cumulative distribution function** (or **c.d.f.**) of $X$ is denoted $\map F X$, and defined as:

- $\forall x \in \R: \map {\map F X} x := \map \Pr {X \le x}$

### CNF

## D

### DCC

### DNF

## E

### EE

Context: Predicate Logic.

- Rule of Existential Elimination, which is another term for the Rule of Existential Instantiation (EI).

### EG

Context: Predicate Logic.

### EI

Context: Predicate Logic.

Alternatively, Rule of Existential Introduction, which is another term for the Rule of Existential Generalisation (EG). Beware.

### EMF

Context: Electromagnetic theory.

## F

### FCF

## G

### GCD or g.c.d.

Also known as:

### GCF

Also known as:

### glb

Another term for **infimum**.

## H

### HCF or h.c.f.

Also known as greatest common divisor (g.c.d.).

## I

### iff

### ICF

### I.V.P.

#### I.V.P.: 1

#### I.V.P.: 2

## J

## K

## L

### LCM, lcm or l.c.m.

### LHS

In an equation:

- $\text {Expression $1$} = \text {Expression $2$}$

the term $\text {Expression $1$}$ is the **left hand side**.

### LSC

## M

## N

### NNF

## O

### ODE

## P

### PCI

### PCFI

### PDE

A **partial differential equation** (abbreviated **P.D.E.** or **PDE**) is a **differential equation** which has:

- one dependent variable
- more than one independent variable.

The derivatives occurring in it are therefore partial.

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X: \Omega \to \R$ be a continuous random variable on $\struct {\Omega, \Sigma, \Pr}$.

Let $\Omega_X = \Img X$, the image of $X$.

Then the **probability density function** of $X$ is the mapping $f_X: \R \to \closedint 0 1$ defined as:

- $\forall x \in \R: \map {f_X} x = \begin {cases} \displaystyle \lim_{\epsilon \mathop \to 0^+} \frac {\map \Pr {x - \frac \epsilon 2 \le X \le x + \frac \epsilon 2} } \epsilon & : x \in \Omega_X \\ 0 & : x \notin \Omega_X \end {cases}$

### PFI

### PGF or p.g.f.

Let $X$ be a discrete random variable whose codomain, $\Omega_X$, is a subset of the natural numbers $\N$.

Let $p_X$ be the probability mass function for $X$.

The **probability generating function for $X$**, denoted $\map {\Pi_X} s$, is the formal power series defined by:

- $\displaystyle \map {\Pi_X} s := \sum_{n \mathop = 0}^\infty \map {p_X} n s^n \in \R \left[\left[{s}\right]\right]$

### PID or pid

A **principal ideal domain** is an integral domain in which every ideal is a principal ideal.

### PMF or p.m.f.

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X: \Omega \to \R$ be a discrete random variable on $\struct {\Omega, \Sigma, \Pr}$.

Then the **(probability) mass function** of $X$ is the (real-valued) function $p_X: \R \to \closedint 0 1$ defined as:

- $\forall x \in \R: \map {p_X} x = \begin{cases} \map \Pr {\set {\omega \in \Omega: \map X \omega = x} } & : x \in \Omega_X \\ 0 & : x \notin \Omega_X \end{cases}$

where $\Omega_X$ is defined as $\Img X$, the image of $X$.

That is, $\map {p_X} x$ is the probability that the discrete random variable $X$ takes the value $x$.

### PMI

### PNT

### Poset

A **partially ordered set** is a relational structure $\left({S, \preceq}\right)$ such that $\preceq$ is a partial ordering.

The **partially ordered set** $\left({S, \preceq}\right)$ is said to be **partially ordered by $\preceq$**.

## Q

### Q.E.D.

The initials of **Quod Erat Demonstrandum**, which is Latin for **which was to be demonstrated**.

These initials were traditionally added to the end of a proof, after the last line which is supposed to contain the statement that was to be proved.

The usage is considered hopelessly old-fashioned nowadays, and is rarely seen outside nerd sitcoms.

The Halmos symbol $\blacksquare$ is usually used instead.

$\mathsf{Pr} \infty \mathsf{fWiki}$ universally uses $\blacksquare$ to signify the end of a proof.

The symbol $\Box$ is used to signify the end of a stage part way through a proof, where a subsidiary result is proved.

## R

### RHS

In an equation:

- $\text {Expression $1$} = \text {Expression $2$}$

the term $\text {Expression $2$}$ is the **right hand side**.

## S

### SCF

### SFCF

### SHM

### SICF

### SUVAT

- A category of problems in elementary applied mathematics and Newtonian physics involving a body $B$ under constant acceleration $\mathbf a$.

- They consist of applications of the equations:

### $(1):$ Velocity after Time

- $\mathbf v = \mathbf u + \mathbf a t$

### $(2):$ Distance after Time

- $\mathbf s = \mathbf u t + \dfrac {\mathbf a t^2} 2$

### $(3):$ Velocity after Distance

- $\mathbf v \cdot \mathbf v = \mathbf u \cdot \mathbf u + 2 \mathbf a \cdot \mathbf s$

where:

- $\mathbf u$ is the velocity at time $t = 0$
- $\mathbf v$ is the velocity at time $t$
- $\mathbf s$ is the displacement of $B$ from its initial position at time $t$
- $\cdot$ denotes the scalar product.

The term **SUVAT** arises from the symbols used, $\mathbf s$, $\mathbf u$, $\mathbf v$, $\mathbf a$ and $t$.

## T

## U

### UF

- Ultrafilter Lemma (also UL).

### UFD

### UL

- Ultrafilter Lemma (also UF).

### URM

- Unlimited register machine: an abstraction of a computing device with certain particular characteristics.

## V

## W

### WFF

### WLOG

Suppose there are several cases which need to be investigated.

If the same argument can be used to dispose of two or more of these cases, then it is acceptable in a proof to pick just one of these cases, and announce this fact with the words: **Without loss of generality, ...**, or just **WLOG**.

### WRT or w.r.t.

When performing calculus operations, that is differentiation or integration, one needs to announce which variable one is "working with".

Thus the phrase **with respect to** is (implicitly or explicitly) part of every statement in calculus.

## X

## Y

## Z

### ZF

### ZFC

- Zermelo-Fraenkel set theory with the Axiom of Choice.