🔰二元關係

集合 ⟩ 關係 ⟩ 二元關係

兩個集合 Ａ、Ｂ (Ａ、Ｂ可以是同一個集合) 的元素之間的關係 ，稱為「二元關係」(binary relation)，事實上就是一種Ａ到Ｂ的「對應關係」，通常以：

• $\mathbf{A} \times \mathbf{B}= \{ \ (a,b) \ | \ a \in \mathbf{A}, b \in \mathbf{B} \}$

的子集 (subset) 來表示。如果我們用 ${\color{orange}\mapsto}$ 來表示此二元關係，則：

• $(a,b) \in {\color{orange}\mapsto}$ 代表 a 對 b 有此關係。

• 此時我們用 $a \ {\color{orange}\mapsto} \ b$ 表示。

(⭐ 注意： $a \ {\color{orange}\mapsto} \ b$ 不代表有 $b \ {\color{orange}\mapsto} \ a$❗)

• 二元關係常用表格來表示 (👉 二元關係屬性)

🔴 主題

• 二元關係

  • equivalence relations: satisfy reflexivity, symmetry, and transitivity. 💠 example： $a \equiv b \text{ (mod 5)}$
  • partial orders: satisfy reflexivity, antisymmetry, and transitivity. 💠 example： $A \subseteq B$ (is a subset of)
  • 全序╱total ordering： satisfy totality, antisymmetry, and transitivity. 💠 example： $x \le y$
  • functions: satisfy a special property called functional dependence. In a function ${\color{orange}f}:A \to B$, each element of $A$ is associated with exactly one element of $B$, that is,

    • $\forall x \in A, \exists {\color{orange}!} \ y \in B \ni (x,y) \in {\color{orange}f}$.
• 二元關係屬性

🔯 符號

參考：KaTeX ⟩ Relations

= =	≑≑ \doteqdot	⪅⪅ \lessapprox	⌣⌣ \smile
<< <	≖≖ \eqcirc	⋚⋚ \lesseqgtr	⊏⊏ \sqsubset
>> >	−: \eqcolon or \minuscolon	⪋⪋ \lesseqqgtr	⊑⊑ \sqsubseteq
:: :	−:: \Eqcolon or \minuscoloncolon	≶≶ \lessgtr	⊐⊐ \sqsupset
≈≈ \approx	=: \eqqcolon or \equalscolon	≲≲ \lesssim	⊒⊒ \sqsupseteq
≈: \approxcolon	=:: \Eqqcolon or \equalscoloncolon	≪≪ \ll	⋐⋐ \Subset
≈::\approxcoloncolon	≂≂ \eqsim	⋘⋘ \lll	⊂⊂ \subset or \sub
≊≊ \approxeq	⪖⪖ \eqslantgtr	⋘⋘ \llless	⊆⊆ \subseteq or \sube
≍≍ \asymp	⪕⪕ \eqslantless	<< \lt	⫅⫅ \subseteqq
∍∍ \backepsilon	≡≡ \equiv	∣∣ \mid	≻≻ \succ
∽∽ \backsim	≒≒ \fallingdotseq	⊨⊨ \models	⪸⪸ \succapprox
⋍⋍ \backsimeq	⌢⌢ \frown	⊸⊸ \multimap	≽≽ \succcurlyeq
≬≬ \between	≥≥ \ge	⊶⊶ \origof	⪰⪰ \succeq
⋈⋈ \bowtie	≥≥ \geq	∋∋ \owns	≿≿ \succsim
≏≏ \bumpeq	≧≧ \geqq	∥∥ \parallel	⋑⋑ \Supset
≎≎ \Bumpeq	⩾⩾ \geqslant	⊥⊥ \perp	⊃⊃ \supset
≗≗ \circeq	≫≫ \gg	⋔⋔ \pitchfork	⊇⊇ \supseteq or \supe
:≈ \colonapprox	⋙⋙ \ggg	≺≺ \prec	⫆⫆ \supseteqq
::≈ \Colonapprox or \coloncolonapprox	⋙⋙ \gggtr	⪷⪷ \precapprox	≈≈ \thickapprox
:− \coloneq or \colonminus	>> \gt	≼≼ \preccurlyeq	∼∼ \thicksim
::− \Coloneq or \coloncolonminus	⪆⪆ \gtrapprox	⪯⪯ \preceq	⊴⊴ \trianglelefteq
:= \coloneqq or \colonequals	⋛⋛ \gtreqless	≾≾ \precsim	≜≜ \triangleq
::= \Coloneqq or \coloncolonequals	⪌⪌ \gtreqqless	∝∝ \propto	⊵⊵ \trianglerighteq
:∼ \colonsim	≷≷ \gtrless	≓≓ \risingdotseq	∝∝ \varpropto
::∼ \Colonsim or \coloncolonsim	≳≳ \gtrsim	∣∣ \shortmid	△△ \vartriangle
≅≅ \cong	⊷⊷ \imageof	∥∥ \shortparallel	⊲⊲ \vartriangleleft
⋞⋞ \curlyeqprec	∈∈ \in or \isin	∼∼ \sim	⊳⊳ \vartriangleright
⋟⋟ \curlyeqsucc	⋈⋈ \Join	∼: \simcolon	: \vcentcolon or \ratio
⊣⊣ \dashv	≤≤ \le	∼::\simcoloncolon	⊢⊢ \vdash
:: \dblcolon or \coloncolon	≤≤ \leq	≃≃ \simeq	⊨⊨ \vDash
≐≐ \doteq	≦≦ \leqq	⌢⌢ \smallfrown	⊩⊩ \Vdash
≑≑ \Doteq	⩽⩽ \leqslant	⌣⌣ \smallsmile	⊪⊪ \Vvdash

Direct Input: =<>:∈∋∝∼∽≂≃≅≈≊≍≎≏≐≑≒≓≖≗≜≡≤≥≦≧≫≬≳≷≺≻≼≽≾≿⊂⊃⊆⊇⊏⊐⊑⊒⊢⊣⊩⊪⊸⋈⋍⋐⋑⋔⋙⋛⋞⋟⌢⌣⩾⪆⪌⪕⪖⪯⪰⪷⪸⫅⫆≲⩽⪅≶⋚⪋⊥⊨⊶⊷=<>:∈∋∝∼∽≂≃≅≈≊≍≎≏≐≑≒≓≖≗≜≡≤≥≦≧≫≬≳≷≺≻≼≽≾≿⊂⊃⊆⊇⊏⊐⊑⊒⊢⊣⊩⊪⊸⋈⋍⋐⋑⋔⋙⋛⋞⋟⌢⌣⩾⪆⪌⪕⪖⪯⪰⪷⪸⫅⫆≲⩽⪅≶⋚⪋⊥⊨⊶⊷ ≔ ≕ ⩴

Negated Relations

≠= \not =

⪊⪊ \gnapprox	≱ \ngeqslant	⊈⊈ \nsubseteq	⪵⪵ \precneqq
⪈⪈ \gneq	≯≯ \ngtr	⊈ \nsubseteqq	⋨⋨ \precnsim
≩≩ \gneqq	≰≰ \nleq	⊁⊁ \nsucc	⊊⊊ \subsetneq
⋧⋧ \gnsim	≰ \nleqq	⋡⋡ \nsucceq	⫋⫋ \subsetneqq
≩ \gvertneqq	≰ \nleqslant	⊉⊉ \nsupseteq	⪺⪺ \succnapprox
⪉⪉ \lnapprox	≮≮ \nless	⊉ \nsupseteqq	⪶⪶ \succneqq
⪇⪇ \lneq	∤∤ \nmid	⋪⋪ \ntriangleleft	⋩⋩ \succnsim
≨≨ \lneqq	∉∈/ \notin	⋬⋬ \ntrianglelefteq	⊋⊋ \supsetneq
⋦⋦ \lnsim	∌∋ \notni	⋫⋫ \ntriangleright	⫌⫌ \supsetneqq
≨ \lvertneqq	∦∦ \nparallel	⋭⋭ \ntrianglerighteq	⊊ \varsubsetneq
≆≆ \ncong	⊀⊀ \nprec	⊬⊬ \nvdash	⫋ \varsubsetneqq
≠= \ne	⋠⋠ \npreceq	⊭⊭ \nvDash	⊋ \varsupsetneq
≠= \neq	∤ \nshortmid	⊯⊯ \nVDash	⫌ \varsupsetneqq
≱≱ \ngeq	∦ \nshortparallel	⊮⊮ \nVdash
≱ \ngeqq	≁≁ \nsim	⪹⪹ \precnapprox

Direct Input: ∉∌∤∦≁≆≠≨≩≮≯≰≱⊀⊁⊈⊉⊊⊋⊬⊭⊮⊯⋠⋡⋦⋧⋨⋩⋬⋭⪇⪈⪉⪊⪵⪶⪹⪺⫋⫌∈/∋∤∦≁≆=≨≩≮≯≰≱⊀⊁⊈⊉⊊⊋⊬⊭⊮⊯⋠⋡⋦⋧⋨⋩⋬⋭⪇⪈⪉⪊⪵⪶⪹⪺⫋⫌

📗 參考

• wiki ⟩

  • Cartesian product (數學「關係」是「笛卡兒積」的子集)

  • adjacency list (也算是一種數學「關係」)