Skip to main content

# 9: Sets

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

• 9.1: Basics
Object: any distinct entity
• 9.2: Defining sets
Remember that mathematical notation is about communicating mathematical information. Since a set is defined by its member objects, to communicate the details of a set of objects one needs to provide a means to decide whether any given object is or is not an element of the set.
• 9.3: Subsets and equality of sets
Often we want to distinguish a collection of certain “special” elements within a larger set of elements.
• 9.4: Complement, union, and intersection
First, it is often convenient to restrict the scope of the discussion.
• 9.5: Cartesian Product
the set of all possible ordered pairs of elements from two given sets A and B, where the first element in a pair is from A and the second is from B
• 9.6: Alphabets and words
any set can be considered an alphabet
• 9.7: Sets of sets
Sets can be made up of any kind of objects, even other sets! (But now we must be careful of the use of the phrase “contained in”.)
• 9.8: Activities
• 9.9: Exercises

This page titled 9: Sets is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Jeremy Sylvestre via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

• Was this article helpful?