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CS1527/notes-2024-02-02.md

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CS1527/notes-2024-02-06.md

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CS1527/notes-2024-02-13.md

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CS1534/notes-2024-02-08.md

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CS1534/notes-2024-02-13.md

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MA1006/decls.tex

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 \newcommand{\abs}[1]{\left|#1\right|}
 \newcommand{\polar}[2]{#1\paren{\cos{\paren{#2}} + i\sin{\paren{#2}}}}
 \newcommand{\adj}[1]{\operatorname{adj}#1}
+\newcommand{\card}[1]{\left|#1\right|}
 
 \makeatletter
 \renewcommand*\env@matrix[1][*\c@MaxMatrixCols c]{%

MA1511/decls.tex

@@ -21,6 +21,7 @@
 \newcommand{\abs}[1]{\left|#1\right|}
 \newcommand{\polar}[2]{#1\paren{\cos{\paren{#2}} + i\sin{\paren{#2}}}}
 \newcommand{\adj}[1]{\operatorname{adj}#1}
+\newcommand{\card}[1]{\left|#1\right|}
 
 \makeatletter
 \renewcommand*\env@matrix[1][*\c@MaxMatrixCols c]{%

MA1511/sets.tex

@@ -2,7 +2,8 @@
 \title{Sets}
 \begin{document}
 \maketitle
-Set comprehensions can be written $\{ x | x \in \N \}$ or $\{ x : x \in \N \}$ - '$:$' or '$|$'
+Set comprehensions can be written $\{ x | x \in \N \}$ or $\{ x : x \in \N \}$ - '$:$' or '$|$' \\
+Sets are defined entirely by the values of $x$ for which $x \in A$
 \begin{description}
 	\item[Axiom of Extensionality / Set Equality] $A = B \iff \forall x. (x \in A \iff x \in B)$
 	\item[$A \subseteq B$] \quad $\forall x \in A. x \in B$ \\
@@ -13,5 +14,46 @@ Set comprehensions can be written $\{ x | x \in \N \}$ or $\{ x : x \in \N \}$ -
 	\item[$\cup$] Union
 	\item[$\cap$] Intersection
 	\item[$A \setminus B$] \quad $\{ x \in A : x \not\in B \}$
+		\item[$A^\complement$]\quad $U \setminus A$
+	\item[$[a, b)$] \quad $\{ x \in \R : a \leq x < b \}$
+\end{description}
+\begin{align*}
+	C \setminus (A \cup B) \equiv           & ~ (C \setminus A) \cap (C \setminus B) \\
+	C \setminus (A \cap B) \equiv           & ~ (C \setminus A) \cup (C \setminus B) \\
+	(A^\complement)^\complement \equiv      & ~ A                                    \\
+	A^\complement \cup B^\complement \equiv & ~ (A \cap B)^\complement               \\
+	A^\complement \cap B^\complement \equiv & ~ (A \cup B)^\complement
+\end{align*}
+\section*{Families of Sets}
+A family of sets indexed by a set $I$ (the indexing set): $A_i ~~\forall~i\in I (\equiv (A_i)_{i\in I})$ \\
+$A_i$ is a set for every element $i \in I$ \\
+A family of sets indexed by $\N$ is called a sequence of sets. Also written $(B_i)^{\inf}_{i=0}$ or $(B_i)_{i \geq 0}$
+\begin{align*}
+\bigcup_{i \in I}~A_i \equiv \{x | \exists i \in I. x \in A_i \} \\
+\bigcap_{i \in I}~A_i \equiv \{x | \forall i \in I. x \in A_i \} & \text{ Exists iff } \exists~i\in I
+\end{align*}
+\begin{align*}
+	& \forall i \in I. A_i \subseteq \cup_{j \in I}A_j \\
+	& \forall i \in I. A_i \subseteq B \implies \cup_{j \in I}A_j \subseteq B \\
+	& \forall i \in I.\cap_{j \in I}A_j \subseteq A_i  \\
+	& \forall i \in I. B \subseteq A_i \implies B \subseteq \cap_{j \in I}A_j \\ \\
+	& B \cup \cap_{i \in I}A_i = \cup_{i \in I}(B \cap A_i) \\
+	& B \cap \cup_{i \in I}A_i = \cap_{i \in I}(B \cup A_i) \\
+	& B \setminus \cup_{i \in I}A_i = \cap_{i\in I}(B\setminus A_i) \\
+	& B \setminus \cap_{i \in I}A_i = \cup_{i\in I}(B \setminus A_i)
+\end{align*}
+\section*{Cartesian Products}
+Ordered pairs can be represented as $(x, y) \equiv \{x, \{x, y\}\}$ \\
+$X \times Y = \{ (x, y) |~ \forall x, y.~x \in X \land y \in Y\}$ for sets $X$, $Y$ \\
+$X^n$ is $X\times X^{n-1}$ for set $X$ and natural $n$ \\
+$\card{X\times Y} = \card{X} \times \card{Y}$ (for finite $X$, $Y$)
+\section*{Functions}
+For sets $X$, $Y$:
+\begin{description}
+	\item[A function $F: X \to Y$] $\subseteq X\times Y$ \text { where } \\
+		$\forall x \in X.~\exists \text{ a unique } y \in Y.~ (x, y) \in F$ \\
+		$F(x)$ denotes the unique element $y \in Y$ for which $(x, F(x)) \in F$
+	\item[$\operatorname{dom}(F)$] The domain of $F$, i.e. $X$
+	\item[$\operatorname{incl}^{X}_{A} : A \to X$] $= a \quad \forall A X. \text{ where } A \subseteq X$
 \end{description}
 \end{document}