# Symbols:O

### 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`

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### 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}`

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### 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}`

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### Set of Octonions

- $\Bbb O$

The $\LaTeX$ code for \(\mathbb O\) is `\mathbb O`

or `\Bbb O`

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### 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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