Definition of Mathematical Symbols

Table of Contents

On this page, some of the mathematical symbols are presented with their meanings, ways of reading, and LaTeX commands. The rest will be added later continuously.

Set theory

$\in, \ni, \notin, \notni$ (set membership)

Set membership represents the relationship between a set $A$ and its element $x$¹ ².

$x \in A$:
- $x$ is an element of the set $A$. $x$ belongs to the set $A$. $x$ is in the set $A$.
$x \ni A$:
- The set $A$ contains $x$ as an element.
$x \notin A$:
- $x$ is not an element of the set $A$.
- This can also be written as $\neg(x \in A)$ (see “$\neg$ logical negation”).

LaTeX commands:

$\in$, \in
$\ni$ , \ni
$\notin$, \notin
$\notni$ , \notni

On the machine learning context, set membership is used like below:

If a sample $x$ in a training dataset $C$ ($C \in x$) takes binary values ($x \in \lbrace 0, 1 \rbrace$), the logistic function is often used as a loss function.

$\subset, \supset, \subseteq, \supseteq, \subsetneq, \supsetneq, \not\subset, \not\supset$ (set inclusion)

Set inclusion represents the relationship between a set $A$ and a set $B$. Note: There are two ways to use these symbols depends on contexts¹ ² ³.

Subset:
- $A \subset B$: $A$ is included in $B$, and possibly is equal to $B$, which means every elements of $A$ is included in $B$.
- This can be written as $\forall A, x \in A \Rightarrow x \in B$ (see “$\forall$ universal quantification”, “$\Rightarrow$ material conditional”).
Proper subset:
- $A \subset B$: $A$ is not equal to $B$, and every elements of $A$ belongs to $B$.
- It is equal to $A \ne B \wedge \forall A, x \in A \Rightarrow x \in B$ (see “$\wedge$ logical and”).

LaTeX commands:

$\subset$, \subset
$\supset$ , \supset
$\subseteq$, \subseteq
$\supseteq$, \supseteq
$\subsetneq$, \subsetneq
$\supsetneq$, \supsetneq
$\not\subset$, \not\subset
$\not\supset$, \not\supset

Basic logic

$\neg$ (logical negation)

$\neg$, \neg

$\forall$ (universal quantification)

$\Rightarrow$ (material conditional)

$\wedge$ (logical and)

Equality, equivalence and similarity

$\approx$ (approximate equal)

e.g. $\pi \approx 3.14159$

LaTeX command:

$\approx$, \approx

$\sim$ (tilde)

Instead of $\approx$ (approximatively equal)
Two numbers have the same order of magnitude as.
$X \sim N(\mu, \sigma^2)$ means the random variable $x$ follows the normal distribution with mean $\mu$ and variance $\sigma^2$.

LaTeX command:

$\sim$, \sim