Added set size draft

Advanced/Size of Sets/parts/0 sets.tex

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+\section{Set Basics}
+
+\definition{}
+A \textit{set} is a collection of objects. \par
+If $a$ is an element of set $S$, we write $a \in S$. This is pronounced \say{$a$ in $S$.} \par
+The position of each element in a set or the number of times it is repeated doesn't matter. \par
+All that matters is \textit{which} elements are in the set.
+
+\vspace{2mm}
+
+We say two sets $A$ and $B$ are equal if every element of $A$ is in $B$, and every element of $B$ is in $A$. This is known as the \textit{principle of extensionality.}
+
+\problem{}
+Convince yourself that $\{a, b\} = \{b, a\} = \{a, b, a, b, b\}$.
+
+\definition{}
+A set $A$ is a \textit{subset} of a set $B$ if every element of $A$ is in $B$. \par
+For example, $\{a, b\}$ is a subset of $\{a, b, c\}$. This is written $\{a, b\} \subseteq \{a, b, c\}$. \par
+Note that the \say{subset} symbol resembles the \say{less than or equal to} symbol.
+
+\vspace{2mm}
+
+We can also write $\{a, b\} \subset \{a, b, c\}$, which denotes a \textit{strict subset.} \par
+The relationship between $\subseteq$ and $\subset$ is the same as the relationship between $\leq$ and $<$. \par
+For example, $\{a, b, c\} \subseteq \{a, b, c\}$ is true, but $\{a, b, c\} \subset \{a, b, c\}$ is false.
+
+
+\definition{}
+The \textit{empty set}, usually written $\varnothing$, is the unique set containing no elements. \par
+\note[Note]{The $\varnothing$ symbol is called \say{varnothing.} If you'd like to know why, ask an instructor.}
+
+\problem{}
+Which of the following are true?
+\begin{itemize}
+	\item $\{1, 3\} = \{3, 3, 1\}$
+	\item $\{1, 2\} \subset \{2\}$
+	\item $\{1, 2\} \subset \{1, 2\}$
+	\item $\{1, 2\} \subseteq \{1, 2\}$
+	\item $\{2\} \subseteq \{1, 2\}$
+	\item $\varnothing \subseteq \{1, 2\}$
+\end{itemize}
+
+\vfill
+\pagebreak
+
+\problem{}
+Let $A$ and $B$ be sets. Convince yourself that $A \subseteq B$ and $B \subseteq A$ implies $A = B$.
+
+\vspace{2mm}
+
+\hint{Whenever you start a proof, you should first look at definitions. \\
+As stated on the previous page, $A = B$ if every element in $A$ is in $B$ and every element of $B$ is in $A$.}
+
+\vspace{2mm}
+
+As we saw before, the $\subseteq$ relation behaves a lot like the $\leq$ relation. \par
+The statement above is very similar to the statement \say{$x \leq y$ and $y \geq x$ implies $x = y$}.
+
+
+\definition{}
+Let $A$ be a set. The \textit{power set} of $A$, written $\mathcal{P}(A)$, is the set of all subsets of $A$.
+
+\problem{}
+What is the power set of $\{1, 2, 3\}$? \par
+\hint{It has eight elements.}
+
+\vfill
+
+\problem{}
+Let $A$ be a set with $n$ elements. \par
+How many elements does $\mathcal{P}(A)$ have? \par
+\hint{Binary may help.}
+
+\vfill
+\pagebreak
+
+
+\definition{Set Operations}
+$A \cap B$ is the \textit{intersection} of $A$ and $B$. \par
+It is the set of objects that are in both $A$ and $B$.
+
+\vspace{3mm}
+
+$A \cup B$ is the \textit{union} of $A$ and $B$. \par
+It is the set of objects that are in either $A$ or $B$.
+
+\vspace{3mm}
+
+$A - B$ is the \textit{difference} of $A$ and $B$. \par
+It is the set of objects that are in $A$ but are not in $B$.
+
+\problem{}
+What is $\{a, b, c\} \cap \{b, c, d\}$?
+
+\vfill
+
+\problem{}
+What is $\{a, b, c\} \cup \{b, c, d\}$?
+
+\vfill
+
+\problem{}
+What is $\{a, b, c\} - \{b, c, d\}$?
+
+\vfill
+\pagebreak