抽象代数笔记 - 等价关系

Wednesday, January 24, 2024

D = {对, 错}, 定义: $R: A \times B \rightarrow D$ 若 $R(a, b)$ = 对, 称为 $a$ 与 $b$ 符合关系 $R$, 记为 $aRb$ 若$R(a, b) = 错$, 称为 $a$ 与 $b$ 不符合关系 $R$.

抽象代数笔记 - 同构映射

Wednesday, January 24, 2024

Def: 同构映射、同构 在 $\phi$ 下 $$ \forall a, b \in A, \forall \bar a, \bar b \in \bar A, (a \mapsto \bar a, b \mapsto \bar b) \Rightarrow (a \circ b \mapsto \bar a \bar \circ \bar b) $$ $A$ 与 $\bar A$ 同构, 记作 $A \cong \bar A$. $$ \phi : x \mapsto x + t \ \phi : x…

Hello World

Tuesday, January 23, 2024

Hello World h2 一一一 二二二 三三三 h3 123 456 789 h4 粗体 斜体 删除线 h5 链接

Nix Language - Data types /etc ./foo.png ~/.config { foo.bar = 1; } { foo = { bar = 1; }; } rec {} Primitives String “hello ${{a=“1”; b=“2”;}.b}” “1 2 ${toString 3}” “${pkgs.bash}/bin/sh” “foo bar” ’' foo bar ’' “foo\nbar\n” string interpolation ${} ‘’${} ‘’\n Number 123 .23e13 Path /bin/sh ./builder.sh (relative to the directory of the Nix expression that contained it) ~/foo at least one slash must appear before any interpolated expresssion.…

Nix Language - Operators String: + Number: + - * / Relation: == != Boolean: ! Attr: a.x a.x or 1 Sets: {} // {} Precedence Pre Asso Name Syntax 1 none Attribute selection attrset . attrpath [ or expr ] 2 left Function application func expr 3 none Arithmetic negation - number 4 none Has attribute attrset ? attrpath 5 right List concatenation list ++ list 6 left Multiplication number * number left Division number / number 7 left Subtraction number - number left Addition number + number left String concatenation string + string left Path concatenation path + path left Path and string concatenation path + string left String and path concatenation string + path 8 none Logical negation (NOT) !…

Nix Language - Expressions Let-expressions let-in: let [ identifier = expr; ]… in expr let x = 1; y = 2; in x + y Inheriting attributes inherit pkgs src; inherit (pkgs) src pkg1 pkg2 pkg3; Adds the variables to the current scope (attribute set or let bindings). let x = { a = 1; b = 2; }; inherit (builtins) attrNames; in { names = attrNames x; } let x = { a = 1; b = 2; }; in { names = builtins.…

Functions pattern: body x: x + 1 x: y: x + y (x: x + 1) 100 let inc = x: x + 1; in inc (inc (inc 100)) { x, y }: x + y { x, y ? “bar” }: x + y { x, y, … }: x + y { x, y } @ args: x + y args @ { x, y }: x + y args@{ x, y, z, … }: z + y + x + args.…

“hello ${{a=“1”; b=“2”;}.b}” “1 2 ${toString 3}” “${pkgs.bash}/bin/sh” Data types /etc ./foo.png ~/.config { foo.bar = 1; } { foo = { bar = 1; }; } rec {} Primitives String “foo bar” ’' foo bar ’' “foo\nbar\n” string interpolation ${} ‘’${} ‘’\n Number 123 .23e13 Path /bin/sh ./builder.sh (relative to the directory of the Nix expression that contained it) ~/foo at least one slash must appear before any interpolated expresssion.…