Relations and Partitions

$\newcommand{\dom}{\operatorname{Dom}} \newcommand{\codom}{\operatorname{Codom}} \newcommand{\range}{\operatorname{Rng}} \newcommand{\lcm}{\operatorname{lcm}} \newcommand{\card}{\operatorname{card}} \newcommand{\rel}{\operatorname{rel}} \newcommand{\Rel}{\operatorname{Rel}} \newcommand{\logeq}{\Longleftrightarrow} \renewcommand{\implies}{\Longrightarrow} \def\tcboxmath{} \def\coloneqq{:=} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} $

Chapter 7 Relations and Partitions

The impediment to action advances action. What stands in the way becomes the way.
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―Marcus Aurelius, Roman emperor
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While there is no agreed upon universal definition of mathematics, one could argue that mathematics focuses on the study of patterns and relationships. Certain types of relationships occur over and over in mathematics. One way of formalizing the abstract nature and structure of these relationships is with the notion of relations. In Chapter 8, we will see that a function is a special type of relation.

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