Mathematical Notations
Contents
Mathematical Notations#
We largely follow the Machine Learning: The Basics book in terms of notations.
Set Notation#
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 \(1,2,\ldots\). |
\(\mathbb{R}\) |
The set of real numbers \(x\). |
\(\mathbb{R}_{+}\) |
The set of non-negative real numbers \(x \geq 0\). |
\(\{0,1\}\) |
The set consisting of two real-number 0 and 1. |
\([0,1]\) |
The closed interval of real numbers \(x\) with \(0 \leq x \leq 1\). |
\(f(\cdot), h(\cdot)\) |
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. |
Notation |
Description |
|---|---|
\(a \in \mathcal{A} \quad\) |
This statement indicates that the object \(a\) is an element of the set \(\mathcal{A}\). |