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

\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
In particular, if $A \subset B$, $A \subseteq B$ and $A \neq B$ \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
By definition, the empty set is a subset of every set. \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}

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\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

\problem{}
Show that the set of all sets that do not contain themselves is not a set. \par


\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