# Symbols:Abbreviations

## A

### AC

- The axiom of choice.

Same as AoC.

### ACC

### a.e.

- Abbreviation for
**almost everywhere**.

### agm

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

### alg.

### AoC

- The axiom of choice.

Same as AC.

### arith.

- Abbreviation for
**arithmetic**or**arithmetical**.

## 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

### cdf or c.d.f.

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

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

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

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

### CNF or cnf

## D

### D.C.

Some sources refer to a **derivative** as a **differential coefficient**, and abbreviate it **D.C.**

Some sources call it a **derived function**.

### DCC

### DNF or dnf

## E

### EE

Context: Predicate Logic.

- Rule of Existential Elimination, which is another term for the Rule of Existential Instantiation ($\text {EI}$).

### EG

Context: Predicate Logic.

### EI

Context: Predicate Logic.

Alternatively, Rule of Existential Introduction, which is another term for the Rule of Existential Generalisation ($\text {EG}$). Beware.

### EMF

Context: Electromagnetism.

**Electromotive force** is a quantity that measures the source of potential energy in an electric circuit.

It is defined as the amount of work per unit electric charge.

## F

### FCF

## G

### GCD or g.c.d.

Also known as:

### GCF

Also known as:

### GCH

- The Generalized Continuum Hypothesis can be abbreviated $\operatorname {GCH}$.

### glb

Another term for **infimum**.

## H

### HCF or h.c.f.

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

### hex

### hyp. $(1)$

### hyp. $(2)$

## 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**.

### LQR

Let $p$ and $q$ be distinct odd primes.

Then:

- $\paren {\dfrac p q} \paren {\dfrac q p} = \paren {-1}^{\dfrac {\paren {p - 1} \paren {q - 1} } 4}$

where $\paren {\dfrac p q}$ and $\paren {\dfrac q p}$ are defined as the Legendre symbol.

An alternative formulation is: $\paren {\dfrac p q} = \begin{cases} \quad \paren {\dfrac q p} & : p \equiv 1 \lor q \equiv 1 \pmod 4 \\ -\paren {\dfrac q p} & : p \equiv q \equiv 3 \pmod 4 \end{cases}$

The fact that these formulations are equivalent is immediate.

This fact is known as the **Law of Quadratic Reciprocity**, or **LQR** for short.

### lsb or l.s.b.

### LSC

## M

### msb or m.s.b.

## 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 $X$ be an absolutely continuous random variable on $\struct {\Omega, \Sigma, \Pr}$.

Let $P_X$ be the probability distribution of $X$.

Let $\map \BB \R$ be the Borel $\sigma$-algebra of $\R$.

Let $\lambda$ be the Lebesgue measure on $\struct {\R, \map \BB \R}$.

We define the **probability density function** $f_X$ by:

- $\ds f_X = \frac {\d P_X} {\d \lambda}$

where $\dfrac {\d P_X} {\d \lambda}$ denotes the Radon-Nikodym derivative of $P_X$ with respect to $\lambda$.

### 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:

- $\ds \map {\Pi_X} s := \sum_{n \mathop = 0}^\infty \map {p_X} n s^n \in \R \sqbrk {\sqbrk s}$

### 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 $\struct {S, \preceq}$ such that $\preceq$ is a partial ordering.

The **partially ordered set** $\struct {S, \preceq}$ is said to be **partially ordered by $\preceq$**.

## Q

### QED

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.

### QEF

The initials of **Quod Erat Faciendum**, which is Latin for **which was to be done**.

These initials were traditionally added to the end of a geometric construction.

### QEI

The initials of **Quod Erat Inveniendum**, which is Latin for **which was to be found**.

These initials were traditionally added after the completion of a calculation (either arithmetic or algebraic).

## R

### RHS

In an equation:

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

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

## S

### SAA

### SAS

### SCF

### SFCF

### SHM

### SICF

### SSS

### 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 dot 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

### Zermelo-Fraenkel Set Theory

**ZF**

An abbreviation for **Zermelo-Fraenkel Set Theory**, a system of axiomatic set theory upon which most of conventional mathematics can be based.

### Zermelo-Fraenkel Set Theory with the Axiom of Choice

**ZFC**

An abbreviation for **Zermelo-Fraenkel Set Theory with the Axiom of Choice**, a system of axiomatic set theory upon which the whole of (at least conventional) mathematics can be based.