Mathematical Notations

We largely follow the Machine Learning: The Basics book in terms of notations.

Set Notation

Table 3 Set Notations

Notation	Description
$a\in \mathcal{A}$	This statement indicates that the object $a$ is an element of the set $\mathcal{A}$.
$a:=b$	This statement defines $a$ to be shorthand for $b$.
$|\mathcal{A}|$	The cardinality (number of elements) of a finite set $\mathcal{A}$.
$\mathcal{A}\subseteq \mathcal{B}$	$\mathcal{A}$ is a subset of $\mathcal{B}$.
$\mathcal{A}\subset \mathcal{B}$	$\mathcal{A}$ is a strict subset of $\mathcal{B}$.
$\mathbb{N}$	The set of natural numbers $1,2,\dots$.
$\mathbb{R}$	The set of real numbers $x$.
${\mathbb{R}}_{+}$	The set of non-negative real numbers $x\ge 0$.
$\{0,1\}$	The set consisting of two real-number 0 and 1.
$[0,1]$	The closed interval of real numbers $x$ with $0\le x\le 1$.
$f(\cdot ),h(\cdot )$	A function or map $f(\cdot )$ that accepts any element $a\in \mathcal{A}$ from a set $\mathcal{A}$ as input and delivers a well-defined element $f(a)\in \mathcal{B}$ of a set $\mathcal{B}$. The set $\mathcal{A}$ is the domain of the function $f$ and the set $\mathcal{B}$ is the codomain of $f$. Machine Learning revolves around finding (or learning) a function $h$ (which we call hypothesis) that reads in the features $\mathbf{x}$ of a data point and delivers a prediction $h(\mathbf{x})$ for the label $y$ of the data point.

Table 4 Linear Algebra Notations

Notation	Description
$a\in \mathcal{A}$	This statement indicates that the object $a$ is an element of the set $\mathcal{A}$.