From 7dedef56658de2a0c912be9fa0b69247287ad585 Mon Sep 17 00:00:00 2001 From: bluepython508 Date: Wed, 14 Feb 2024 20:59:32 +0000 Subject: [PATCH] . --- CS1527/notes-2024-02-02.md | 1 + CS1527/notes-2024-02-06.md | 1 + CS1527/notes-2024-02-13.md | 1 + CS1534/notes-2024-02-08.md | 1 + CS1534/notes-2024-02-13.md | 1 + MA1006/decls.tex | 1 + MA1511/decls.tex | 1 + MA1511/sets.tex | 44 +++++++++++++++++++++++++++++++++++++- 8 files changed, 50 insertions(+), 1 deletion(-) create mode 100644 CS1527/notes-2024-02-02.md create mode 100644 CS1527/notes-2024-02-06.md create mode 100644 CS1527/notes-2024-02-13.md create mode 100644 CS1534/notes-2024-02-08.md create mode 100644 CS1534/notes-2024-02-13.md diff --git a/CS1527/notes-2024-02-02.md b/CS1527/notes-2024-02-02.md new file mode 100644 index 0000000..8b13789 --- /dev/null +++ b/CS1527/notes-2024-02-02.md @@ -0,0 +1 @@ + diff --git a/CS1527/notes-2024-02-06.md b/CS1527/notes-2024-02-06.md new file mode 100644 index 0000000..8b13789 --- /dev/null +++ b/CS1527/notes-2024-02-06.md @@ -0,0 +1 @@ + diff --git a/CS1527/notes-2024-02-13.md b/CS1527/notes-2024-02-13.md new file mode 100644 index 0000000..8b13789 --- /dev/null +++ b/CS1527/notes-2024-02-13.md @@ -0,0 +1 @@ + diff --git a/CS1534/notes-2024-02-08.md b/CS1534/notes-2024-02-08.md new file mode 100644 index 0000000..8b13789 --- /dev/null +++ b/CS1534/notes-2024-02-08.md @@ -0,0 +1 @@ + diff --git a/CS1534/notes-2024-02-13.md b/CS1534/notes-2024-02-13.md new file mode 100644 index 0000000..8b13789 --- /dev/null +++ b/CS1534/notes-2024-02-13.md @@ -0,0 +1 @@ + diff --git a/MA1006/decls.tex b/MA1006/decls.tex index e738aea..f36f46b 100644 --- a/MA1006/decls.tex +++ b/MA1006/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]{% diff --git a/MA1511/decls.tex b/MA1511/decls.tex index e738aea..f36f46b 100644 --- a/MA1511/decls.tex +++ b/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]{% diff --git a/MA1511/sets.tex b/MA1511/sets.tex index 95f5fc4..b695350 100644 --- a/MA1511/sets.tex +++ b/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}