diff --git a/.gitignore b/.gitignore index 48c4576..39bea9c 100644 --- a/.gitignore +++ b/.gitignore @@ -1,3 +1,4 @@ *.pdf latex.out/ .direnv +__pycache__/ diff --git a/CS1534/notes-2024-02-15.md b/CS1534/notes-2024-02-15.md new file mode 100644 index 0000000..8b13789 --- /dev/null +++ b/CS1534/notes-2024-02-15.md @@ -0,0 +1 @@ + diff --git a/MA1006/decls.tex b/MA1006/decls.tex index f36f46b..86a52d7 100644 --- a/MA1006/decls.tex +++ b/MA1006/decls.tex @@ -22,9 +22,16 @@ \newcommand{\polar}[2]{#1\paren{\cos{\paren{#2}} + i\sin{\paren{#2}}}} \newcommand{\adj}[1]{\operatorname{adj}#1} \newcommand{\card}[1]{\left|#1\right|} +\newcommand{\littletaller}{\mathchoice{\vphantom{\big|}}{}{}{}} +\newcommand{\restr}[2]{{% we make the whole thing an ordinary symbol + \left.\kern-\nulldelimiterspace % automatically resize the bar with \right + #1 % the function + \littletaller % pretend it's a little taller at normal size + \right|_{#2} % this is the delimiter + }} \makeatletter -\renewcommand*\env@matrix[1][*\c@MaxMatrixCols c]{% +\renewcommand*{\env@matrix}[1][*\c@MaxMatrixCols c]{% \hskip -\arraycolsep \let\@ifnextchar\new@ifnextchar \array{#1}} diff --git a/MA1511/decls.tex b/MA1511/decls.tex index f36f46b..86a52d7 100644 --- a/MA1511/decls.tex +++ b/MA1511/decls.tex @@ -22,9 +22,16 @@ \newcommand{\polar}[2]{#1\paren{\cos{\paren{#2}} + i\sin{\paren{#2}}}} \newcommand{\adj}[1]{\operatorname{adj}#1} \newcommand{\card}[1]{\left|#1\right|} +\newcommand{\littletaller}{\mathchoice{\vphantom{\big|}}{}{}{}} +\newcommand{\restr}[2]{{% we make the whole thing an ordinary symbol + \left.\kern-\nulldelimiterspace % automatically resize the bar with \right + #1 % the function + \littletaller % pretend it's a little taller at normal size + \right|_{#2} % this is the delimiter + }} \makeatletter -\renewcommand*\env@matrix[1][*\c@MaxMatrixCols c]{% +\renewcommand*{\env@matrix}[1][*\c@MaxMatrixCols c]{% \hskip -\arraycolsep \let\@ifnextchar\new@ifnextchar \array{#1}} diff --git a/MA1511/sets.tex b/MA1511/sets.tex index b695350..3d2f467 100644 --- a/MA1511/sets.tex +++ b/MA1511/sets.tex @@ -29,18 +29,18 @@ A family of sets indexed by a set $I$ (the indexing set): $A_i ~~\forall~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 + \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) + & \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 $(x, y) \equiv \{x, \{x, y\}\}$ \\ @@ -51,9 +51,50 @@ $\card{X\times Y} = \card{X} \times \card{Y}$ (for finite $X$, $Y$) 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 $(x, F(x)) \in F$ + $\forall x \in X.~\exists \text{ a unique } F(x) \in Y.~ (x, F(x)) \in F$ \\ + There exist $\card{Y}^{\card{X}}$ functions $F: X \to Y$ \item[$\operatorname{dom}(F)$] The domain of $F$, i.e. $X$ - \item[$\operatorname{incl}^{X}_{A} : A \to X$] $= a \quad \forall A X. \text{ where } A \subseteq X$ + \item[$\operatorname{incl}^X_A : A \to X$] $= a \quad \forall A,X. \text{ where } A \subseteq X$ + \item[Constant function] $\exists y_0 \in Y.~\forall x \in X.~ f(x) = y_0$ + \item[Characteristic Function of a set $A \subseteq X$: $\chi_A: X \to \{0, 1\}$] + \[ + \chi_A: X \to \{0, 1\} = \left\{\begin{array}{lr} + 0 & \text{ if } x \not\in A \\ + 1 & \text{ if } x \in A + \end{array} \right. + \] + \item[Restriction of a function $f: X \to Y$] $\restr{f}{A}$ is $f$ specialized contravariantly to $A \subseteq X$ + \item[$f(A)$: Image of $A$ under $f$] $f$ mapped over $A$ \quad for function $f: X \to Y$, $A \subseteq X$ + \item[$\operatorname{ran}(f)$ / image of $f$ / range of $f$] $\{ f(x) | x \in X \}$, $f(X)$, i.e. all possible values of $f(x)$ + \item[Preimage of $B$ under $f$] $\{ x \in X ~|~ f(x) \in B \}$ \\ + written $f^{-1}(B)$, but is \emph{not} the inverse of f +\end{description} +For $f: X \to Y$, $A \subseteq A' \subseteq X$, $B \subseteq B' \subseteq Y$: +\begin{align*} + f(A) \subseteq~& f(A') \\ + f^{-1}(B) \subseteq ~& f^{-1}(B') \\ + f^{-1}(f(A)) \supseteq ~& A \\ + f(f^{-1}(B)) \subseteq ~& B +\end{align*} +For set families $(A_i \subseteq X)_{i \in I}, (B_j \subseteq Y)_{j \in J}$: +\begin{align*} + f(\cup_{i \in I} A_i) = ~& \cup_{i \in I}f(A_i) \\ + f(\cap_{i \in I} A_i) \subseteq ~& \cap_{i \in I}f(A_i) \\ + f^{-1}(\cup_{j \in J} B_j) = ~& \cup_{j \in J} f^{-1}(B_j) \\ + f^{-1}(\cap_{j \in J} B_j) = ~& \cap_{j \in J} f^{-1}(B_j) +\end{align*} +\section*{Function Composition} +For functions $f: X \to Y$, $g: Y \to Z$: +\begin{align*} + & (g \circ f): X \to Z \\ + & (g\circ f)(x) = g(f(x)) & \forall x \in X \\ \\ +\end{align*} +\section*{Surjection and Injection} +For $f: X \to Y$, $f$ is +\begin{description} + \item[surjective] iff $f(X) = Y$, i.e. $\forall y \in Y.~ \exists x \in X.~ f(x) = y$ \\ + Range is codomain, 'onto' + \item[injective] iff $\forall x, x' \in X. ~ f(x) = f(x') \implies x = x'$ + \item[bijective] iff $f$ is injective and $f$ is surjective \end{description} \end{document}