80 lines
2.9 KiB
TeX
80 lines
2.9 KiB
TeX
% File: 20-01-2022.tex
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% Created: 12:28:35 Thu, 20 Jan 2022 EST
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% Last Change: 12:28:35 Thu, 20 Jan 2022 EST
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%
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\documentclass[letterpaper]{article}
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\usepackage{amsmath}
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\usepackage{graphicx}
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\usepackage{cancel}
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\usepackage{amssymb}
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\usepackage{listings}
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\usepackage[shortlabels]{enumitem}
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\usepackage{soul}
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%\usepackage[smartEllipses,hashEnumerators,hybrid]{markdown}
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\usepackage{geometry}
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\usepackage{dirtytalk}
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\usepackage{lplfitch}
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\geometry{portrait, margin=1in}
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%\begin{minted}[linenos,bgcolor=LightGray]{[language]}
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\date{01/20/2022}
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\title{%
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Intro Set Theory\\
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\large MATH--190--01 : Discrete Mathematics for Computing}
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\author{Blizzard MacDougall}
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\begin{document}
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\maketitle
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\pagenumbering{arabic}
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\section{Sets}
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A set is an unordered list. A set is said to contain elements or objects.
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Notationally, $a\in S$ is \say{a is contained in S}. $a\notin S$ also exists.
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A set is typically an upper case letter, and lower case letters are elements.
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The roster method just lists all the elements in a list, with curly braces around them.
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If you're going to use an ellipses, you need to give enough info for it to be obvious.
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Sets don't \emph{have} to have any common properties, but its helpful if they do.
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There's also \emph{set builder notation}. That is, $S={x\in D | P(x)}$, read as
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\say{The set $S$, defined as all $x$ contained in $D$ such that $P(x)$}.
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Examples:
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\begin{equation}
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\begin{split}
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\mathbb{N}=\{0,1,2,3,...\}\\
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\mathbb{Z}=\{...-3,-2,-1,0,1,2,3,...\}\\
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\mathbb{Z}^+=\mathbb{Z}_+=\mathbb{N}+0\\
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\mathbb{Q}=\{\frac{a}{b}|a,b\in \mathbb{Z}\ and\ b\neq0\}\\
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\mathbb{R}=All\ real\ numbers\\
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\mathbb{R}^+=\mathbb{R}_+=\{x\in \mathbb{R}| x> 0\}\\
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\mathbb{C}=all\ complex\ numbers
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\end{split}
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\end{equation}
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Note that natural numbers are kinda fuzzy. This course says zero is natural, others may not.
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Every set is a subset of itself.
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Subsets are noted as $A\subseteq B$. i.e. A is a subset of B.
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\begin{equation}
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\mathbb{N}\subseteq\mathbb{Z}\subseteq\mathbb{R}\subseteq\mathbb{Q}\subseteq\mathbb{C}
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\end{equation}
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Every set has two difinitive subsets, itself, and the empty set (or the null set).$\emptyset$
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A subset that is strictly smaller than the superset is called a proper subset.
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You can check for this with $\subset$.
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Note that there is only one empty set.
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Two sets are equal if they contain the same elements. Alternatively, sets are equal
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if they contain each other. To show something's unique, show that if there are two of the thing,
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then prove they're equal.
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\section{Cardinality}
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Note that we're sticking with finite sets for now.
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If we have a set with exactly $n$ distinct elements, (where $n\in \mathbb{N}$), then
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the cardinality of a set is $n$. Cardinality is notationally written as $\#S=|S|=n$.
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If $S$ is not finite, its infinite. Cardinality of infinte sets are written with $\aleph_n$.
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Notably, $|\emptyset|=0$.
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$\wp$
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\end{document}
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