# Symbols:Abbreviations

## A

### AC

The axiom of choice.

Same as AoC.

### ACC

The ascending chain condition.

### a.e.

Abbreviation for almost everywhere.

### agm

Abbreviation for arithmetic-geometric mean.

### alg.

Abbreviation for algebra or algebraic.

### 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

The Boolean Prime Ideal Theorem.

## C

### cdf or c.d.f.

Cumulative distribution function.

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

Conjunctive normal form.

## D

### D.C.

Differential Coefficient:

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

Some sources call it a derived function.

### DCC

Descending chain condition

### DNF or dnf

Disjunctive normal form.

## 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.

Rule of Existential Generalisation.

### EI

Context: Predicate Logic.

Rule of Existential Instantiation.

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

### EMF

Context: Electromagnetism.

Electromotive force.

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

Finite continued fraction.

## G

### GCD or g.c.d.

Greatest common divisor.

Also known as:

highest common factor (hcf or h.c.f.)
greatest common factor (gcf or g.c.f.)

### GCF

Greatest common divisor.

Also known as:

greatest common factor (GCD, gcd or g.c.d.)
highest common factor (HCF, hcf or h.c.f.)

### GCH

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

### glb

Greatest lower bound.

Another term for infimum.

## H

### HCF or h.c.f.

Highest common factor.

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

Hypotenuse.

Hypothesis.

## I

If and only if.

### ICF

Infinite continued fraction.

### I.V.P.

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

The intermediate value property.

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

An initial value problem.

## L

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

The lowest (or least) common multiple.

### LHS

Left hand side.

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.

Least significant bit.

### LSC

Lower semicontinuous.

## M

### msb or m.s.b.

Most significant bit.

## N

### NNF

Negation normal form.

## O

### ODE

Ordinary differential equation.

## P

### PCI

Principle of Complete Induction (or Principle of Complete Finite Induction).

### PCFI

Principle of Complete Finite Induction.

### 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.

### pdf

Probability density function:

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

Principle of Finite Induction.

### PGF or p.g.f.

Probability generating function:

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

Principal ideal domain:

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

### PMF or p.m.f.

Probability mass function:

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

Principle of Mathematical Induction.

### PNT

Prime Number Theorem.

### Poset

Partially Ordered Set.

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

Quod Erat Demonstrandum

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

Quod Erat Faciendum

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

Quod Erat Inveniendum

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

Right hand side.

In an equation:

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

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

## S

### SAA

Triangle Side-Angle-Angle Equality.

### SAS

Triangle Side-Angle-Side Equality.

### SCF

Simple continued fraction.

### SFCF

Simple finite continued fraction.

### SHM

Simple harmonic motion.

### SICF

Simple infinite continued fraction.

### SSS

Triangle Side-Side-Side Equality.

### 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 $t$.

## U

### UF

Ultrafilter Lemma (also UL).

### UFD

Unique factorization domain.

### UL

Ultrafilter Lemma (also UF).

### URM

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

## W

### WFF

Well-formed formula.

### WLOG

Without loss of generality.

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.

(Differentiation) with respect to.

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.

## 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.