#### 4.3 Mathematical Notation

4.3.1 Common Sets
4.3.2 Set Notation
4.3.3 Math-Operators

In this section we are going to defines some mathematical notation that is used in the document.

##### 4.3.1 Common Sets

The common sets of natural numbers, integers, rationals, reals, and complex numbers are defined below. Furthermore we define some notation specific to combinatorial species.

25mathematics 25  (20)  26
\newcommand{\setFont}{\protect\mathbb}
\newcommand{\fieldFont}{\protect\mathbb}
\newcommand{\setC}{\setFont{C}} % complex numbers
\newcommand{\setN}{\setFont{N}} % natural numbers
\newcommand{\setQ}{\setFont{Q}} % rational numbers
\newcommand{\setR}{\setFont{R}} % real numbers
\newcommand{\setZ}{\setFont{Z}} % integers

\newcommand{\fieldK}{\fieldFont{K}}

\newcommand{\category}{\protect\mathbb}
\newcommand{\categoryFiniteSet}{\category{E}}
\newcommand{\categoryE}{\categoryFiniteSet} % abbreviation
\newcommand{\categoryBijectiveFiniteSet}{\category{B}}
\newcommand{\categoryB}{\categoryBijectiveFiniteSet} %abbreviation

Uses Q 47 and Z 47.

##### 4.3.2 Set Notation

In order to get the size of braces correct in set definitions, we provide the following commands.

26mathematics 25+   (20)  25  28a
\newcommand{\Set}[1]{\left\{#1\right\}}
\newcommand{\setDef}[2]{{#1}\left|\,\vphantom{#1}{#2}\right.}
\newcommand{\SetDef}[2]{\Set{\,\setDef{#1}{#2}\,}}

Then we have

• $\Set{a,b,c}$
• $\Set{\Set{a,b},c}$
• $\Set{\Set{a,b},c}$

• $\Set{a,\frac{b}{c}}$
• $\Set{a,\frac{b}{c}}$

• $\SetDef{\int_{a}^\infty f(x) dx}{a\in\Set{3,4,5}}$
• $\SetDef{\int_{a}^\infty f(x) dx}{a\in\Set{3,4,5}}$

The power set of a set X is denoted by P(X).

28amathematics 25+   (20)  26  28b
\newcommand{\PowerSet}[1]{\ensuremath{\protect\mathcal{P}(#1)}}

With U we denote the restriction of some function f to a subset U of its original source domain.

28bmathematics 25+   (20)  28a  28c
\newcommand{\restrictedTo}[2]{\left.#1\right|_{#2}}

##### 4.3.3 Math-Operators

Not all mathematical operators are defined in the amsmath package, so we define here a few more.

28cmathematics 25+   (20)  28b  28d
\DeclareMathOperator{\card}{card}  % set cardinality
\DeclareMathOperator{\ord}{ord}  % order of a series
\DeclareMathOperator{\supp}{supp}  % support of a polynomial
\DeclareMathOperator{\fix}{fix}  % number of fixed points
\DeclareMathOperator{\Fix}{Fix}  % the fixed elements
\DeclareMathOperator{\aut}{aut}  % aut k = 1^{k_1}k_1! ... n^{k_n}k_n!s
\DeclareMathOperator{\lcm}{lcm}  % least common multiple

##### 4.3.4 Functorial Composition

The following definition is used for denoting the functorial composite F G of two species F and G.

28dmathematics 25+   (20)  28c  29
\def\functorialcompose{\mathbin{\rlap{$\sqcap$}\sqcup}}

##### 4.3.5 Theorem-like Environments
29mathematics 25+   (20)  28d
\usepackage{amsthm}
\theoremstyle{plain}
\newtheorem{Theorem}{Theorem}[section]
\newtheorem{Lemma}[Theorem]{Lemma}
\newtheorem{Corollary}[Theorem]{Corollary}
\newtheorem{Example}[Theorem]{Example}
\newtheorem{Proposition}[Theorem]{Proposition}

% Can have an argument in square brackets (Proof name).
\newenvironment{Proof}{\begin{proof}}{\end{proof}}

\newtheorem{Problem}{Problem}

\theoremstyle{remark}
\newtheorem{Remark}[Theorem]{Remark}

\theoremstyle{definition}
\newtheorem{Definition}[Theorem]{Definition}

Uses name 198.