# Symbols:O

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### Falsehood

$0$

Symbol often used in the context of computer science for falsehood.

A statement has a truth value of false if and only if what it says does not match the way that things are.

The $\LaTeX$ code for $0$ is 0 .

### Big-O Notation

$\OO$

Used for example as follows in the context of sequences:

Let $g: \N \to \R$ be a real sequence, expressed here as a real-valued function on the set of natural numbers $\N$.

Then $\map \OO g$ is defined as:

$\map \OO g = \set {f: \N \to \R: \exists c \in \R_{>0}: \exists n_0 \in \N: \forall n > n_0: 0 \le \size {\map f n} \le c \cdot \size {\map g n} }$

The $\LaTeX$ code for $a_n = \map \OO {b_n}$ is a_n = \map \OO {b_n} .

### Little-O Notation

$o$

Used for example as follows in the context of sequences:

Let $g: \N \to \R$ be a real sequence, expressed here as a real-valued function on the set of natural numbers $\N$.

Then $\map \oo g$ is defined as:

$\map \oo g = \set {f: \N \to \R: \forall c \in \R_{>0}: \exists n_0 \in \N: \forall n > n_0: \size {\map f n} \le c \cdot \size {\map g n} }$

This is denoted:

$a_n = \map o {b_n}$

The $\LaTeX$ code for $a_n = \map o {b_n}$ is a_n = \map o {b_n} .

### Set of Octonions

$\Bbb O$

The set of octonions.

The $\LaTeX$ code for $\mathbb O$ is \mathbb O  or \Bbb O.

### Order Type

$\ot$

Let $\struct {S, \preccurlyeq_1}$ and $\struct {T, \preccurlyeq_2}$ be ordered sets.

Then $S$ and $T$ have the same (order) type if and only if they are order isomorphic.

The order type of an ordered set $\struct {S, \preccurlyeq}$ can be denoted $\map \ot {S, \preccurlyeq}$.

The $\LaTeX$ code for $\map \ot {S, \preccurlyeq}$ is \map \ot {S, \preccurlyeq} .

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