Case mapping

There are three sets of characters: upper case, lower case, and neither. toUpperCase() and toLowerCase() provide mappings between them. Let's define the following sets:

• $\mathbb{C}$ is the set of all single-code-point Unicode characters.
• $I$ := $\{ c\in\mathbb{C}\mid \mathtt{toUpperCase}(c) = \mathtt{toLowerCase}(c) = c \}$ are characters that are invariant under both toUpperCase() and toLowerCase(), such as numbers and emojis. These are "uninteresting" characters that aren't in the scope of discussion.
  • $I$ is closed under both functions, i.e. there does not exist a character $c\in\mathbb{C}$ such that $c\notin I$ but $\mathtt{toUpperCase}(c)\in I$ or $\mathtt{toLowerCase}(c)\in I$. This has been built into the data collector code shown below, so you can check that there are no logs in the browser console. (Note that $I$ also includes code points that aren't assigned to characters, so the exact set of characters is hard to get.)
• $\mathbb{UC}$ := $\mathbb{C}\setminus I$. Characters in $\mathbb{UC}$ are never mapped to $I$. Size: 2907
• $M_L$ := $\{ c\in\mathbb{UC}\mid \mathtt{toLowerCase}(\mathtt{toUpperCase}(c))\in\mathbb{C}, \mathtt{toUpperCase}(c)\notin\mathbb{C} \}$ =
  Character set (21)

  • ǰ (U+01F0) → J̌ (U+004A U+030C) → ǰ (U+01F0)
  • ΐ (U+0390) → Ϊ́ (U+03AA U+0301) → ΐ (U+0390)
  • ΰ (U+03B0) → Ϋ́ (U+03AB U+0301) → ΰ (U+03B0)
  • ẖ (U+1E96) → H̱ (U+0048 U+0331) → ẖ (U+1E96)
  • ẗ (U+1E97) → T̈ (U+0054 U+0308) → ẗ (U+1E97)
  • ẘ (U+1E98) → W̊ (U+0057 U+030A) → ẘ (U+1E98)
  • ẙ (U+1E99) → Y̊ (U+0059 U+030A) → ẙ (U+1E99)
  • ὐ (U+1F50) → Υ̓ (U+03A5 U+0313) → ὐ (U+1F50)
  • ὒ (U+1F52) → Υ̓̀ (U+03A5 U+0313 U+0300) → ὒ (U+1F52)
  • ὔ (U+1F54) → Υ̓́ (U+03A5 U+0313 U+0301) → ὔ (U+1F54)
  • ὖ (U+1F56) → Υ̓͂ (U+03A5 U+0313 U+0342) → ὖ (U+1F56)
  • ᾶ (U+1FB6) → Α͂ (U+0391 U+0342) → ᾶ (U+1FB6)
  • ῆ (U+1FC6) → Η͂ (U+0397 U+0342) → ῆ (U+1FC6)
  • ῒ (U+1FD2) → Ϊ̀ (U+03AA U+0300) → ῒ (U+1FD2)
  • ῖ (U+1FD6) → Ι͂ (U+0399 U+0342) → ῖ (U+1FD6)
  • ῗ (U+1FD7) → Ϊ͂ (U+03AA U+0342) → ῗ (U+1FD7)
  • ῢ (U+1FE2) → Ϋ̀ (U+03AB U+0300) → ῢ (U+1FE2)
  • ῤ (U+1FE4) → Ρ̓ (U+03A1 U+0313) → ῤ (U+1FE4)
  • ῦ (U+1FE6) → Υ͂ (U+03A5 U+0342) → ῦ (U+1FE6)
  • ῧ (U+1FE7) → Ϋ͂ (U+03AB U+0342) → ῧ (U+1FE7)
  • ῶ (U+1FF6) → Ω͂ (U+03A9 U+0342) → ῶ (U+1FF6)
  To summarize, they are:
  • Latin Extended-B
    • LATIN SMALL LETTER J WITH CARON (U+01F0)
  • Greek and Coptic
    • GREEK SMALL LETTER IOTA WITH DIALYTIKA AND TONOS (U+0390)
    • GREEK SMALL LETTER UPSILON WITH DIALYTIKA AND TONOS (U+03B0)
  • Latin Extended Additional
    • 4 characters
  • Greek Extended
    • 14 characters
• $M_U$ := $\{ c\in\mathbb{UC}\mid \mathtt{toUpperCase}(\mathtt{toLowerCase}(c))\in\mathbb{C}, \mathtt{toLowerCase}(c)\notin\mathbb{C} \}$ =
  Character set (1)

  • İ (U+0130) → i̇ (U+0069 U+0307) → İ (U+0130)
  The only character is:
  • Latin Extended-A
    • LATIN CAPITAL LETTER I WITH DOT ABOVE (U+0130)
• $N_L$ := $\{ c\in\mathbb{UC}\mid \mathtt{toLowerCase}(\mathtt{toUpperCase}(c))\notin\mathbb{C}, \mathtt{toUpperCase}(c)\notin\mathbb{C} \}$ =
  Character set (79)

  • ß (U+00DF) → SS (U+0053 U+0053) → ss (U+0073 U+0073)
  • ʼn (U+0149) → ʼN (U+02BC U+004E) → ʼn (U+02BC U+006E)
  • և (U+0587) → ԵՒ (U+0535 U+0552) → եւ (U+0565 U+0582)
  • ẚ (U+1E9A) → Aʾ (U+0041 U+02BE) → aʾ (U+0061 U+02BE)
  • ᾀ (U+1F80) → ἈΙ (U+1F08 U+0399) → ἀι (U+1F00 U+03B9)
  • ᾁ (U+1F81) → ἉΙ (U+1F09 U+0399) → ἁι (U+1F01 U+03B9)
  • ᾂ (U+1F82) → ἊΙ (U+1F0A U+0399) → ἂι (U+1F02 U+03B9)
  • ᾃ (U+1F83) → ἋΙ (U+1F0B U+0399) → ἃι (U+1F03 U+03B9)
  • ᾄ (U+1F84) → ἌΙ (U+1F0C U+0399) → ἄι (U+1F04 U+03B9)
  • ᾅ (U+1F85) → ἍΙ (U+1F0D U+0399) → ἅι (U+1F05 U+03B9)
  • ᾆ (U+1F86) → ἎΙ (U+1F0E U+0399) → ἆι (U+1F06 U+03B9)
  • ᾇ (U+1F87) → ἏΙ (U+1F0F U+0399) → ἇι (U+1F07 U+03B9)
  • ᾈ (U+1F88) → ἈΙ (U+1F08 U+0399) → ἀι (U+1F00 U+03B9)
  • ᾉ (U+1F89) → ἉΙ (U+1F09 U+0399) → ἁι (U+1F01 U+03B9)
  • ᾊ (U+1F8A) → ἊΙ (U+1F0A U+0399) → ἂι (U+1F02 U+03B9)
  • ᾋ (U+1F8B) → ἋΙ (U+1F0B U+0399) → ἃι (U+1F03 U+03B9)
  • ᾌ (U+1F8C) → ἌΙ (U+1F0C U+0399) → ἄι (U+1F04 U+03B9)
  • ᾍ (U+1F8D) → ἍΙ (U+1F0D U+0399) → ἅι (U+1F05 U+03B9)
  • ᾎ (U+1F8E) → ἎΙ (U+1F0E U+0399) → ἆι (U+1F06 U+03B9)
  • ᾏ (U+1F8F) → ἏΙ (U+1F0F U+0399) → ἇι (U+1F07 U+03B9)
  • ᾐ (U+1F90) → ἨΙ (U+1F28 U+0399) → ἠι (U+1F20 U+03B9)
  • ᾑ (U+1F91) → ἩΙ (U+1F29 U+0399) → ἡι (U+1F21 U+03B9)
  • ᾒ (U+1F92) → ἪΙ (U+1F2A U+0399) → ἢι (U+1F22 U+03B9)
  • ᾓ (U+1F93) → ἫΙ (U+1F2B U+0399) → ἣι (U+1F23 U+03B9)
  • ᾔ (U+1F94) → ἬΙ (U+1F2C U+0399) → ἤι (U+1F24 U+03B9)
  • ᾕ (U+1F95) → ἭΙ (U+1F2D U+0399) → ἥι (U+1F25 U+03B9)
  • ᾖ (U+1F96) → ἮΙ (U+1F2E U+0399) → ἦι (U+1F26 U+03B9)
  • ᾗ (U+1F97) → ἯΙ (U+1F2F U+0399) → ἧι (U+1F27 U+03B9)
  • ᾘ (U+1F98) → ἨΙ (U+1F28 U+0399) → ἠι (U+1F20 U+03B9)
  • ᾙ (U+1F99) → ἩΙ (U+1F29 U+0399) → ἡι (U+1F21 U+03B9)
  • ᾚ (U+1F9A) → ἪΙ (U+1F2A U+0399) → ἢι (U+1F22 U+03B9)
  • ᾛ (U+1F9B) → ἫΙ (U+1F2B U+0399) → ἣι (U+1F23 U+03B9)
  • ᾜ (U+1F9C) → ἬΙ (U+1F2C U+0399) → ἤι (U+1F24 U+03B9)
  • ᾝ (U+1F9D) → ἭΙ (U+1F2D U+0399) → ἥι (U+1F25 U+03B9)
  • ᾞ (U+1F9E) → ἮΙ (U+1F2E U+0399) → ἦι (U+1F26 U+03B9)
  • ᾟ (U+1F9F) → ἯΙ (U+1F2F U+0399) → ἧι (U+1F27 U+03B9)
  • ᾠ (U+1FA0) → ὨΙ (U+1F68 U+0399) → ὠι (U+1F60 U+03B9)
  • ᾡ (U+1FA1) → ὩΙ (U+1F69 U+0399) → ὡι (U+1F61 U+03B9)
  • ᾢ (U+1FA2) → ὪΙ (U+1F6A U+0399) → ὢι (U+1F62 U+03B9)
  • ᾣ (U+1FA3) → ὫΙ (U+1F6B U+0399) → ὣι (U+1F63 U+03B9)
  • ᾤ (U+1FA4) → ὬΙ (U+1F6C U+0399) → ὤι (U+1F64 U+03B9)
  • ᾥ (U+1FA5) → ὭΙ (U+1F6D U+0399) → ὥι (U+1F65 U+03B9)
  • ᾦ (U+1FA6) → ὮΙ (U+1F6E U+0399) → ὦι (U+1F66 U+03B9)
  • ᾧ (U+1FA7) → ὯΙ (U+1F6F U+0399) → ὧι (U+1F67 U+03B9)
  • ᾨ (U+1FA8) → ὨΙ (U+1F68 U+0399) → ὠι (U+1F60 U+03B9)
  • ᾩ (U+1FA9) → ὩΙ (U+1F69 U+0399) → ὡι (U+1F61 U+03B9)
  • ᾪ (U+1FAA) → ὪΙ (U+1F6A U+0399) → ὢι (U+1F62 U+03B9)
  • ᾫ (U+1FAB) → ὫΙ (U+1F6B U+0399) → ὣι (U+1F63 U+03B9)
  • ᾬ (U+1FAC) → ὬΙ (U+1F6C U+0399) → ὤι (U+1F64 U+03B9)
  • ᾭ (U+1FAD) → ὭΙ (U+1F6D U+0399) → ὥι (U+1F65 U+03B9)
  • ᾮ (U+1FAE) → ὮΙ (U+1F6E U+0399) → ὦι (U+1F66 U+03B9)
  • ᾯ (U+1FAF) → ὯΙ (U+1F6F U+0399) → ὧι (U+1F67 U+03B9)
  • ᾲ (U+1FB2) → ᾺΙ (U+1FBA U+0399) → ὰι (U+1F70 U+03B9)
  • ᾳ (U+1FB3) → ΑΙ (U+0391 U+0399) → αι (U+03B1 U+03B9)
  • ᾴ (U+1FB4) → ΆΙ (U+0386 U+0399) → άι (U+03AC U+03B9)
  • ᾷ (U+1FB7) → Α͂Ι (U+0391 U+0342 U+0399) → ᾶι (U+1FB6 U+03B9)
  • ᾼ (U+1FBC) → ΑΙ (U+0391 U+0399) → αι (U+03B1 U+03B9)
  • ῂ (U+1FC2) → ῊΙ (U+1FCA U+0399) → ὴι (U+1F74 U+03B9)
  • ῃ (U+1FC3) → ΗΙ (U+0397 U+0399) → ηι (U+03B7 U+03B9)
  • ῄ (U+1FC4) → ΉΙ (U+0389 U+0399) → ήι (U+03AE U+03B9)
  • ῇ (U+1FC7) → Η͂Ι (U+0397 U+0342 U+0399) → ῆι (U+1FC6 U+03B9)
  • ῌ (U+1FCC) → ΗΙ (U+0397 U+0399) → ηι (U+03B7 U+03B9)
  • ῲ (U+1FF2) → ῺΙ (U+1FFA U+0399) → ὼι (U+1F7C U+03B9)
  • ῳ (U+1FF3) → ΩΙ (U+03A9 U+0399) → ωι (U+03C9 U+03B9)
  • ῴ (U+1FF4) → ΏΙ (U+038F U+0399) → ώι (U+03CE U+03B9)
  • ῷ (U+1FF7) → Ω͂Ι (U+03A9 U+0342 U+0399) → ῶι (U+1FF6 U+03B9)
  • ῼ (U+1FFC) → ΩΙ (U+03A9 U+0399) → ωι (U+03C9 U+03B9)
  • ff (U+FB00) → FF (U+0046 U+0046) → ff (U+0066 U+0066)
  • fi (U+FB01) → FI (U+0046 U+0049) → fi (U+0066 U+0069)
  • fl (U+FB02) → FL (U+0046 U+004C) → fl (U+0066 U+006C)
  • ffi (U+FB03) → FFI (U+0046 U+0046 U+0049) → ffi (U+0066 U+0066 U+0069)
  • ffl (U+FB04) → FFL (U+0046 U+0046 U+004C) → ffl (U+0066 U+0066 U+006C)
  • ſt (U+FB05) → ST (U+0053 U+0054) → st (U+0073 U+0074)
  • st (U+FB06) → ST (U+0053 U+0054) → st (U+0073 U+0074)
  • ﬓ (U+FB13) → ՄՆ (U+0544 U+0546) → մն (U+0574 U+0576)
  • ﬔ (U+FB14) → ՄԵ (U+0544 U+0535) → մե (U+0574 U+0565)
  • ﬕ (U+FB15) → ՄԻ (U+0544 U+053B) → մի (U+0574 U+056B)
  • ﬖ (U+FB16) → ՎՆ (U+054E U+0546) → վն (U+057E U+0576)
  • ﬗ (U+FB17) → ՄԽ (U+0544 U+053D) → մխ (U+0574 U+056D)
  To summarize, they are:
  • Latin-1 Supplement
    • LATIN SMALL LETTER SHARP S (U+00DF)
  • Latin Extended-A
    • LATIN SMALL LETTER N PRECEDED BY APOSTROPHE (U+0149)
  • Latin Extended Additional
    • LATIN SMALL LETTER A WITH RIGHT HALF RING (U+1E9A)
  • Armenian
    • ARMENIAN SMALL LIGATURE ECH YIWN (U+0587)
  • Greek Extended
    • Greek small letters with ypogegrammeni ($Gr_L$): U+1F{8,9,A}0 – U+1F{8,9,A}7, U+1F{B,C,F}2 – U+1F{B,C,F}4, U+1F{B,C,F}7 (the Iota subscript, COMBINING GREEK YPOGEGRAMMENI (U+0345) itself, maps to GREEK CAPITAL LETTER IOTA (U+0399) which will be discussed later)
    • Greek capital letters with prosgegrammeni ($Gr_U$): U+1F{8,9,A}8 – U+1F{8,9,A}F, U+1F{B,C,F}C
  • Alphabetic Presentation Forms
    • All latin ligatures: U+FB00 – U+FB06
    • All Armenian ligatures: U+FB13 – U+FB17
• $N_U$ := $\{ c\in\mathbb{UC}\mid \mathtt{toUpperCase}(\mathtt{toLowerCase}(c))\notin\mathbb{C}, \mathtt{toLowerCase}(c)\notin\mathbb{C} \}$ = ∅
• $M_L'$ := $\mathtt{toUpperCase}(M_L)$. Conceptually, these are uppercase letters that just don't have a single Unicode code point.
• $M_U'$ := $\mathtt{toLowerCase}(M_U)$. Conceptually, these are lowercase letters that just don't have a single Unicode code point.
• $N_L'$ := $\mathtt{toUpperCase}(N_L)$
• $N_U'$ := $\mathtt{toLowerCase}(N_U)$
• $\mathcal{R}_U$ := $\mathtt{toUpperCase}(\mathbb{UC})$. Size: 1464
• $\mathcal{R}_L$ := $\mathtt{toLowerCase}(\mathbb{UC})$. Size: 1485

Also define the following predicates:

• $\mathtt{isUpperCase}(c) = \mathtt{/\backslash p\{Uppercase\_Letter\}/u.test(c)}$
• $\mathtt{isLowerCase}(c) = \mathtt{/\backslash p\{Lowercase\_Letter\}/u.test(c)}$

Thus define the following sets:

• $U_{ext}$ := $\{ c\in\mathbb{C}\mid \mathtt{isUpperCase}(c) \}$. Size: 1876 (Note: Unicode utility lists 1831, with the extra 10 possibly being duplicates by normalization.)
• $L_{ext}$ := $\{ c\in\mathbb{C}\mid \mathtt{isLowerCase}(c) \}$. Size: 2273 (Note: Unicode utility lists 2233, with the extra 10 possibly being duplicates by normalization.)
• $U$ := $U_{ext}\setminus I$. Size: 1350
• $L$ := $L_{ext}\setminus I$. Size: 1441

Define the following terminologies:

• $c$ is upper case if $c\in U_{ext}$.
• $c$ is lower case if $c\in L_{ext}$.
  • upper case and lower case are mutually exclusive: $U_{ext}\cap L_{ext}$ = ∅
• $c$ is cased if $c\in U_{ext}\cup L_{ext}$.
• $c$ is uncased if $c\notin U_{ext}\cup L_{ext}$.
• $c$ is lowercase variant if $\mathtt{toLowerCase}(c)\neq c$.
• $c$ is uppercase variant if $\mathtt{toUpperCase}(c)\neq c$.
• $c$ is case-mapping variant if $c$ is either lowercase variant or uppercase variant.
• $c$ is case-mapping invariant if $c$ is neither lowercase variant nor uppercase variant.

export function toLowerCase(char: string): string;
export function toLowerCase(char: Iterable<string>): Set<string>;
export function toLowerCase(
  char: string | Iterable<string>,
): string | Set<string> {
  if (typeof char === "string") return char.toLowerCase().normalize("NFC");
  return new Set([...char].map(toLowerCase));
}

export function toUpperCase(char: string): string;
export function toUpperCase(char: Iterable<string>): Set<string>;
export function toUpperCase(
  char: string | Iterable<string>,
): string | Set<string> {
  if (typeof char === "string") return char.toUpperCase().normalize("NFC");
  return new Set([...char].map(toUpperCase));
}

export function isChar(char: string): boolean {
  return [...char].length === 1;
}

export function isUpperCase(char: string): boolean {
  return /\p{Uppercase_Letter}/u.test(char);
}

export function isLowerCase(char: string): boolean {
  return /\p{Lowercase_Letter}/u.test(char);
}

export const UC = new Set<string>();
export const U = new Set<string>();
export const L = new Set<string>();
export const Uext = new Set<string>();
export const Lext = new Set<string>();
export const ML = new Set<string>();
export const MU = new Set<string>();
export const ML_ = new Set<string>();
export const MU_ = new Set<string>();
export const NL = new Set<string>();
export const NU = new Set<string>();
export const NL_ = new Set<string>();
export const NU_ = new Set<string>();

export function isI(c: string): boolean {
  return toUpperCase(c) === c && toLowerCase(c) === c;
}

for (let c = 0; c < 0x10ffff; c++) {
  const char = String.fromCodePoint(c).normalize("NFC");
  if (isUpperCase(char)) Uext.add(char);
  if (isLowerCase(char)) Lext.add(char);
  if (!isChar(char) || isI(char)) continue;
  UC.add(char);
  const upper = toUpperCase(char);
  const lower = toLowerCase(char);
  if (isI(upper))
    console.error(`Invariant broken: toUpperCase(${char}) = ${upper} ∈ I`);
  else if (isI(lower))
    console.error(`Invariant broken: toLowerCase(${char}) = ${lower} ∈ I`);
  if (!isChar(upper)) {
    if (isChar(toLowerCase(upper))) {
      ML.add(char);
      ML_.add(upper);
    } else {
      NL.add(char);
      NL_.add(upper);
    }
  }
  if (!isChar(lower)) {
    if (isChar(toUpperCase(lower))) {
      MU.add(char);
      MU_.add(lower);
    } else {
      NU.add(char);
      NU_.add(lower);
    }
  }
  if (isUpperCase(char)) U.add(char);
  if (isLowerCase(char)) L.add(char);
}

export const GrU = new Set(NL.values().filter((c) => toLowerCase(c) !== c));

export const RU = toUpperCase(UC);
export const RL = toLowerCase(UC);

NOTE: To maximize the number of single-code-point characters in discussion, we normalize the output with .normalize("NFC").

Case-mapping properties

Idempotence

The first invariant we want to establish is toUpperCase(toUpperCase(c)) == toUpperCase(c) and toLowerCase(toLowerCase(c)) == toLowerCase(c) for all $c\in\mathbb{UC}$.

• $\{ c\in\mathbb{UC}\mid \mathtt{toUpperCase}(\mathtt{toUpperCase}(c))\neq \mathtt{toUpperCase}(c) \}$ = ∅
• $\{ c\in\mathbb{UC}\mid \mathtt{toLowerCase}(\mathtt{toLowerCase}(c))\neq \mathtt{toLowerCase}(c) \}$ = ∅

This also means that if $c$ is uppercase variant, then $c$ will not be the output of toUpperCase(); similarly, if $c$ is lowercase variant, then $c$ will not be the output of toLowerCase().

Complementary ranges

The ranges of toUpperCase() and toLowerCase() are disjoint:

• $\mathcal{R}_U\cap \mathcal{R}_L$ = ∅

But, they are not partitions of $\mathbb{UC}$:

• $\mathbb{UC}\setminus(\mathcal{R}_U\cup \mathcal{R}_L)$ =
  Character set (31)

  Dž (U+01C5)Lj (U+01C8)Nj (U+01CB)Dz (U+01F2)ᾈ (U+1F88)ᾉ (U+1F89)ᾊ (U+1F8A)ᾋ (U+1F8B)ᾌ (U+1F8C)ᾍ (U+1F8D)ᾎ (U+1F8E)ᾏ (U+1F8F)ᾘ (U+1F98)ᾙ (U+1F99)ᾚ (U+1F9A)ᾛ (U+1F9B)ᾜ (U+1F9C)ᾝ (U+1F9D)ᾞ (U+1F9E)ᾟ (U+1F9F)ᾨ (U+1FA8)ᾩ (U+1FA9)ᾪ (U+1FAA)ᾫ (U+1FAB)ᾬ (U+1FAC)ᾭ (U+1FAD)ᾮ (U+1FAE)ᾯ (U+1FAF)ᾼ (U+1FBC)ῌ (U+1FCC)ῼ (U+1FFC)

27 of these characters are $Gr_U$. The other 4 are:

• Latin Extended-B
  • Ligatures ($La$): U+01C5, U+01C8, U+01CB, U+01F2

These characters cannot be produced by toUpperCase() or toLowerCase() with any input, including themselves.

Relationships between case-mapping variance and case

Does upper(lower) case imply upper(lower)case invariance?

Yes. toUpperCase() and toLowerCase() are identity functions on $U_{ext}$ and $L_{ext}$, respectively.

• $\{ c\in U_{ext}\mid \mathtt{toUpperCase}(c)\neq c \}$ = ∅
• $\{ c\in L_{ext}\mid \mathtt{toLowerCase}(c)\neq c \}$ = ∅

This means upper case implies uppercase invariance, and lower case implies lowercase invariance.

Does upper(lower) case imply lower(upper)case variance?

No. $U_{ext}$ and $L_{ext}$ are not proper subsets of $\mathbb{UC}$:

• $U_{ext}\setminus\mathbb{U}$ =
  Character set (526)

  ϒ (U+03D2)ϓ (U+03D3)ϔ (U+03D4)Ᲊ (U+1C89)ℂ (U+2102)ℇ (U+2107)ℋ (U+210B)ℌ (U+210C)ℍ (U+210D)ℐ (U+2110)ℑ (U+2111)ℒ (U+2112)ℕ (U+2115)ℙ (U+2119)ℚ (U+211A)ℛ (U+211B)ℜ (U+211C)ℝ (U+211D)ℤ (U+2124)ℨ (U+2128)ℬ (U+212C)ℭ (U+212D)ℰ (U+2130)ℱ (U+2131)ℳ (U+2133)ℾ (U+213E)ℿ (U+213F)ⅅ (U+2145)Ɤ (U+A7CB)Ꟍ (U+A7CC)꟎ (U+A7CE)꟒ (U+A7D2)꟔ (U+A7D4)Ꟛ (U+A7DA)Ƛ (U+A7DC)𐵐 (U+10D50)𐵑 (U+10D51)𐵒 (U+10D52)𐵓 (U+10D53)𐵔 (U+10D54)𐵕 (U+10D55)𐵖 (U+10D56)𐵗 (U+10D57)𐵘 (U+10D58)𐵙 (U+10D59)𐵚 (U+10D5A)𐵛 (U+10D5B)𐵜 (U+10D5C)𐵝 (U+10D5D)𐵞 (U+10D5E)𐵟 (U+10D5F)𐵠 (U+10D60)𐵡 (U+10D61)𐵢 (U+10D62)𐵣 (U+10D63)𐵤 (U+10D64)𐵥 (U+10D65)𖺠 (U+16EA0)𖺡 (U+16EA1)𖺢 (U+16EA2)𖺣 (U+16EA3)𖺤 (U+16EA4)𖺥 (U+16EA5)𖺦 (U+16EA6)𖺧 (U+16EA7)𖺨 (U+16EA8)𖺩 (U+16EA9)𖺪 (U+16EAA)𖺫 (U+16EAB)𖺬 (U+16EAC)𖺭 (U+16EAD)𖺮 (U+16EAE)𖺯 (U+16EAF)𖺰 (U+16EB0)𖺱 (U+16EB1)𖺲 (U+16EB2)𖺳 (U+16EB3)𖺴 (U+16EB4)𖺵 (U+16EB5)𖺶 (U+16EB6)𖺷 (U+16EB7)𖺸 (U+16EB8)𝐀 (U+1D400)𝐁 (U+1D401)𝐂 (U+1D402)𝐃 (U+1D403)𝐄 (U+1D404)𝐅 (U+1D405)𝐆 (U+1D406)𝐇 (U+1D407)𝐈 (U+1D408)𝐉 (U+1D409)𝐊 (U+1D40A)𝐋 (U+1D40B)𝐌 (U+1D40C)𝐍 (U+1D40D)𝐎 (U+1D40E)𝐏 (U+1D40F)𝐐 (U+1D410)𝐑 (U+1D411)𝐒 (U+1D412)𝐓 (U+1D413)𝐔 (U+1D414)𝐕 (U+1D415)𝐖 (U+1D416)𝐗 (U+1D417)𝐘 (U+1D418)𝐙 (U+1D419)𝐴 (U+1D434)𝐵 (U+1D435)𝐶 (U+1D436)𝐷 (U+1D437)𝐸 (U+1D438)𝐹 (U+1D439)𝐺 (U+1D43A)𝐻 (U+1D43B)𝐼 (U+1D43C)𝐽 (U+1D43D)𝐾 (U+1D43E)𝐿 (U+1D43F)𝑀 (U+1D440)𝑁 (U+1D441)𝑂 (U+1D442)𝑃 (U+1D443)𝑄 (U+1D444)𝑅 (U+1D445)𝑆 (U+1D446)𝑇 (U+1D447)𝑈 (U+1D448)𝑉 (U+1D449)𝑊 (U+1D44A)𝑋 (U+1D44B)𝑌 (U+1D44C)𝑍 (U+1D44D)𝑨 (U+1D468)𝑩 (U+1D469)𝑪 (U+1D46A)𝑫 (U+1D46B)𝑬 (U+1D46C)𝑭 (U+1D46D)𝑮 (U+1D46E)𝑯 (U+1D46F)𝑰 (U+1D470)𝑱 (U+1D471)𝑲 (U+1D472)𝑳 (U+1D473)𝑴 (U+1D474)𝑵 (U+1D475)𝑶 (U+1D476)𝑷 (U+1D477)𝑸 (U+1D478)𝑹 (U+1D479)𝑺 (U+1D47A)𝑻 (U+1D47B)𝑼 (U+1D47C)𝑽 (U+1D47D)𝑾 (U+1D47E)𝑿 (U+1D47F)𝒀 (U+1D480)𝒁 (U+1D481)𝒜 (U+1D49C)𝒞 (U+1D49E)𝒟 (U+1D49F)𝒢 (U+1D4A2)𝒥 (U+1D4A5)𝒦 (U+1D4A6)𝒩 (U+1D4A9)𝒪 (U+1D4AA)𝒫 (U+1D4AB)𝒬 (U+1D4AC)𝒮 (U+1D4AE)𝒯 (U+1D4AF)𝒰 (U+1D4B0)𝒱 (U+1D4B1)𝒲 (U+1D4B2)𝒳 (U+1D4B3)𝒴 (U+1D4B4)𝒵 (U+1D4B5)𝓐 (U+1D4D0)𝓑 (U+1D4D1)𝓒 (U+1D4D2)𝓓 (U+1D4D3)𝓔 (U+1D4D4)𝓕 (U+1D4D5)𝓖 (U+1D4D6)𝓗 (U+1D4D7)𝓘 (U+1D4D8)𝓙 (U+1D4D9)𝓚 (U+1D4DA)𝓛 (U+1D4DB)𝓜 (U+1D4DC)𝓝 (U+1D4DD)𝓞 (U+1D4DE)𝓟 (U+1D4DF)𝓠 (U+1D4E0)𝓡 (U+1D4E1)𝓢 (U+1D4E2)𝓣 (U+1D4E3)𝓤 (U+1D4E4)𝓥 (U+1D4E5)𝓦 (U+1D4E6)𝓧 (U+1D4E7)𝓨 (U+1D4E8)𝓩 (U+1D4E9)𝔄 (U+1D504)𝔅 (U+1D505)𝔇 (U+1D507)𝔈 (U+1D508)𝔉 (U+1D509)𝔊 (U+1D50A)𝔍 (U+1D50D)𝔎 (U+1D50E)𝔏 (U+1D50F)𝔐 (U+1D510)𝔑 (U+1D511)𝔒 (U+1D512)𝔓 (U+1D513)𝔔 (U+1D514)𝔖 (U+1D516)𝔗 (U+1D517)𝔘 (U+1D518)𝔙 (U+1D519)𝔚 (U+1D51A)𝔛 (U+1D51B)𝔜 (U+1D51C)𝔸 (U+1D538)𝔹 (U+1D539)𝔻 (U+1D53B)𝔼 (U+1D53C)𝔽 (U+1D53D)𝔾 (U+1D53E)𝕀 (U+1D540)𝕁 (U+1D541)𝕂 (U+1D542)𝕃 (U+1D543)𝕄 (U+1D544)𝕆 (U+1D546)𝕊 (U+1D54A)𝕋 (U+1D54B)𝕌 (U+1D54C)𝕍 (U+1D54D)𝕎 (U+1D54E)𝕏 (U+1D54F)𝕐 (U+1D550)𝕬 (U+1D56C)𝕭 (U+1D56D)𝕮 (U+1D56E)𝕯 (U+1D56F)𝕰 (U+1D570)𝕱 (U+1D571)𝕲 (U+1D572)𝕳 (U+1D573)𝕴 (U+1D574)𝕵 (U+1D575)𝕶 (U+1D576)𝕷 (U+1D577)𝕸 (U+1D578)𝕹 (U+1D579)𝕺 (U+1D57A)𝕻 (U+1D57B)𝕼 (U+1D57C)𝕽 (U+1D57D)𝕾 (U+1D57E)𝕿 (U+1D57F)𝖀 (U+1D580)𝖁 (U+1D581)𝖂 (U+1D582)𝖃 (U+1D583)𝖄 (U+1D584)𝖅 (U+1D585)𝖠 (U+1D5A0)𝖡 (U+1D5A1)𝖢 (U+1D5A2)𝖣 (U+1D5A3)𝖤 (U+1D5A4)𝖥 (U+1D5A5)𝖦 (U+1D5A6)𝖧 (U+1D5A7)𝖨 (U+1D5A8)𝖩 (U+1D5A9)𝖪 (U+1D5AA)𝖫 (U+1D5AB)𝖬 (U+1D5AC)𝖭 (U+1D5AD)𝖮 (U+1D5AE)𝖯 (U+1D5AF)𝖰 (U+1D5B0)𝖱 (U+1D5B1)𝖲 (U+1D5B2)𝖳 (U+1D5B3)𝖴 (U+1D5B4)𝖵 (U+1D5B5)𝖶 (U+1D5B6)𝖷 (U+1D5B7)𝖸 (U+1D5B8)𝖹 (U+1D5B9)𝗔 (U+1D5D4)𝗕 (U+1D5D5)𝗖 (U+1D5D6)𝗗 (U+1D5D7)𝗘 (U+1D5D8)𝗙 (U+1D5D9)𝗚 (U+1D5DA)𝗛 (U+1D5DB)𝗜 (U+1D5DC)𝗝 (U+1D5DD)𝗞 (U+1D5DE)𝗟 (U+1D5DF)𝗠 (U+1D5E0)𝗡 (U+1D5E1)𝗢 (U+1D5E2)𝗣 (U+1D5E3)𝗤 (U+1D5E4)𝗥 (U+1D5E5)𝗦 (U+1D5E6)𝗧 (U+1D5E7)𝗨 (U+1D5E8)𝗩 (U+1D5E9)𝗪 (U+1D5EA)𝗫 (U+1D5EB)𝗬 (U+1D5EC)𝗭 (U+1D5ED)𝘈 (U+1D608)𝘉 (U+1D609)𝘊 (U+1D60A)𝘋 (U+1D60B)𝘌 (U+1D60C)𝘍 (U+1D60D)𝘎 (U+1D60E)𝘏 (U+1D60F)𝘐 (U+1D610)𝘑 (U+1D611)𝘒 (U+1D612)𝘓 (U+1D613)𝘔 (U+1D614)𝘕 (U+1D615)𝘖 (U+1D616)𝘗 (U+1D617)𝘘 (U+1D618)𝘙 (U+1D619)𝘚 (U+1D61A)𝘛 (U+1D61B)𝘜 (U+1D61C)𝘝 (U+1D61D)𝘞 (U+1D61E)𝘟 (U+1D61F)𝘠 (U+1D620)𝘡 (U+1D621)𝘼 (U+1D63C)𝘽 (U+1D63D)𝘾 (U+1D63E)𝘿 (U+1D63F)𝙀 (U+1D640)𝙁 (U+1D641)𝙂 (U+1D642)𝙃 (U+1D643)𝙄 (U+1D644)𝙅 (U+1D645)𝙆 (U+1D646)𝙇 (U+1D647)𝙈 (U+1D648)𝙉 (U+1D649)𝙊 (U+1D64A)𝙋 (U+1D64B)𝙌 (U+1D64C)𝙍 (U+1D64D)𝙎 (U+1D64E)𝙏 (U+1D64F)𝙐 (U+1D650)𝙑 (U+1D651)𝙒 (U+1D652)𝙓 (U+1D653)𝙔 (U+1D654)𝙕 (U+1D655)𝙰 (U+1D670)𝙱 (U+1D671)𝙲 (U+1D672)𝙳 (U+1D673)𝙴 (U+1D674)𝙵 (U+1D675)𝙶 (U+1D676)𝙷 (U+1D677)𝙸 (U+1D678)𝙹 (U+1D679)𝙺 (U+1D67A)𝙻 (U+1D67B)𝙼 (U+1D67C)𝙽 (U+1D67D)𝙾 (U+1D67E)𝙿 (U+1D67F)𝚀 (U+1D680)𝚁 (U+1D681)𝚂 (U+1D682)𝚃 (U+1D683)𝚄 (U+1D684)𝚅 (U+1D685)𝚆 (U+1D686)𝚇 (U+1D687)𝚈 (U+1D688)𝚉 (U+1D689)𝚨 (U+1D6A8)𝚩 (U+1D6A9)𝚪 (U+1D6AA)𝚫 (U+1D6AB)𝚬 (U+1D6AC)𝚭 (U+1D6AD)𝚮 (U+1D6AE)𝚯 (U+1D6AF)𝚰 (U+1D6B0)𝚱 (U+1D6B1)𝚲 (U+1D6B2)𝚳 (U+1D6B3)𝚴 (U+1D6B4)𝚵 (U+1D6B5)𝚶 (U+1D6B6)𝚷 (U+1D6B7)𝚸 (U+1D6B8)𝚹 (U+1D6B9)𝚺 (U+1D6BA)𝚻 (U+1D6BB)𝚼 (U+1D6BC)𝚽 (U+1D6BD)𝚾 (U+1D6BE)𝚿 (U+1D6BF)𝛀 (U+1D6C0)𝛢 (U+1D6E2)𝛣 (U+1D6E3)𝛤 (U+1D6E4)𝛥 (U+1D6E5)𝛦 (U+1D6E6)𝛧 (U+1D6E7)𝛨 (U+1D6E8)𝛩 (U+1D6E9)𝛪 (U+1D6EA)𝛫 (U+1D6EB)𝛬 (U+1D6EC)𝛭 (U+1D6ED)𝛮 (U+1D6EE)𝛯 (U+1D6EF)𝛰 (U+1D6F0)𝛱 (U+1D6F1)𝛲 (U+1D6F2)𝛳 (U+1D6F3)𝛴 (U+1D6F4)𝛵 (U+1D6F5)𝛶 (U+1D6F6)𝛷 (U+1D6F7)𝛸 (U+1D6F8)𝛹 (U+1D6F9)𝛺 (U+1D6FA)𝜜 (U+1D71C)𝜝 (U+1D71D)𝜞 (U+1D71E)𝜟 (U+1D71F)𝜠 (U+1D720)𝜡 (U+1D721)𝜢 (U+1D722)𝜣 (U+1D723)𝜤 (U+1D724)𝜥 (U+1D725)𝜦 (U+1D726)𝜧 (U+1D727)𝜨 (U+1D728)𝜩 (U+1D729)𝜪 (U+1D72A)𝜫 (U+1D72B)𝜬 (U+1D72C)𝜭 (U+1D72D)𝜮 (U+1D72E)𝜯 (U+1D72F)𝜰 (U+1D730)𝜱 (U+1D731)𝜲 (U+1D732)𝜳 (U+1D733)𝜴 (U+1D734)𝝖 (U+1D756)𝝗 (U+1D757)𝝘 (U+1D758)𝝙 (U+1D759)𝝚 (U+1D75A)𝝛 (U+1D75B)𝝜 (U+1D75C)𝝝 (U+1D75D)𝝞 (U+1D75E)𝝟 (U+1D75F)𝝠 (U+1D760)𝝡 (U+1D761)𝝢 (U+1D762)𝝣 (U+1D763)𝝤 (U+1D764)𝝥 (U+1D765)𝝦 (U+1D766)𝝧 (U+1D767)𝝨 (U+1D768)𝝩 (U+1D769)𝝪 (U+1D76A)𝝫 (U+1D76B)𝝬 (U+1D76C)𝝭 (U+1D76D)𝝮 (U+1D76E)𝞐 (U+1D790)𝞑 (U+1D791)𝞒 (U+1D792)𝞓 (U+1D793)𝞔 (U+1D794)𝞕 (U+1D795)𝞖 (U+1D796)𝞗 (U+1D797)𝞘 (U+1D798)𝞙 (U+1D799)𝞚 (U+1D79A)𝞛 (U+1D79B)𝞜 (U+1D79C)𝞝 (U+1D79D)𝞞 (U+1D79E)𝞟 (U+1D79F)𝞠 (U+1D7A0)𝞡 (U+1D7A1)𝞢 (U+1D7A2)𝞣 (U+1D7A3)𝞤 (U+1D7A4)𝞥 (U+1D7A5)𝞦 (U+1D7A6)𝞧 (U+1D7A7)𝞨 (U+1D7A8)𝟊 (U+1D7CA)
• $L_{ext}\setminus\mathbb{U}$ =
  Character set (832)

  ĸ (U+0138)ƍ (U+018D)ƛ (U+019B)ƪ (U+01AA)ƫ (U+01AB)ƺ (U+01BA)ƾ (U+01BE)ȡ (U+0221)ȴ (U+0234)ȵ (U+0235)ȶ (U+0236)ȷ (U+0237)ȸ (U+0238)ȹ (U+0239)ɕ (U+0255)ɘ (U+0258)ɚ (U+025A)ɝ (U+025D)ɞ (U+025E)ɟ (U+025F)ɢ (U+0262)ɤ (U+0264)ɧ (U+0267)ɭ (U+026D)ɮ (U+026E)ɰ (U+0270)ɳ (U+0273)ɴ (U+0274)ɶ (U+0276)ɷ (U+0277)ɸ (U+0278)ɹ (U+0279)ɺ (U+027A)ɻ (U+027B)ɼ (U+027C)ɾ (U+027E)ɿ (U+027F)ʁ (U+0281)ʄ (U+0284)ʅ (U+0285)ʆ (U+0286)ʍ (U+028D)ʎ (U+028E)ʏ (U+028F)ʐ (U+0290)ʑ (U+0291)ʓ (U+0293)ʖ (U+0296)ʗ (U+0297)ʘ (U+0298)ʙ (U+0299)ʚ (U+029A)ʛ (U+029B)ʜ (U+029C)ʟ (U+029F)ʠ (U+02A0)ʡ (U+02A1)ʢ (U+02A2)ʣ (U+02A3)ʤ (U+02A4)ʥ (U+02A5)ʦ (U+02A6)ʧ (U+02A7)ʨ (U+02A8)ʩ (U+02A9)ʪ (U+02AA)ʫ (U+02AB)ʬ (U+02AC)ʭ (U+02AD)ʮ (U+02AE)ʯ (U+02AF)ϼ (U+03FC)ՠ (U+0560)ֈ (U+0588)ᲊ (U+1C8A)ᴀ (U+1D00)ᴁ (U+1D01)ᴂ (U+1D02)ᴃ (U+1D03)ᴄ (U+1D04)ᴅ (U+1D05)ᴆ (U+1D06)ᴇ (U+1D07)ᴈ (U+1D08)ᴉ (U+1D09)ᴊ (U+1D0A)ᴋ (U+1D0B)ᴌ (U+1D0C)ᴍ (U+1D0D)ᴎ (U+1D0E)ᴏ (U+1D0F)ᴐ (U+1D10)ᴑ (U+1D11)ᴒ (U+1D12)ᴓ (U+1D13)ᴔ (U+1D14)ᴕ (U+1D15)ᴖ (U+1D16)ᴗ (U+1D17)ᴘ (U+1D18)ᴙ (U+1D19)ᴚ (U+1D1A)ᴛ (U+1D1B)ᴜ (U+1D1C)ᴝ (U+1D1D)ᴞ (U+1D1E)ᴟ (U+1D1F)ᴠ (U+1D20)ᴡ (U+1D21)ᴢ (U+1D22)ᴣ (U+1D23)ᴤ (U+1D24)ᴥ (U+1D25)ᴦ (U+1D26)ᴧ (U+1D27)ᴨ (U+1D28)ᴩ (U+1D29)ᴪ (U+1D2A)ᴫ (U+1D2B)ᵫ (U+1D6B)ᵬ (U+1D6C)ᵭ (U+1D6D)ᵮ (U+1D6E)ᵯ (U+1D6F)ᵰ (U+1D70)ᵱ (U+1D71)ᵲ (U+1D72)ᵳ (U+1D73)ᵴ (U+1D74)ᵵ (U+1D75)ᵶ (U+1D76)ᵷ (U+1D77)ᵺ (U+1D7A)ᵻ (U+1D7B)ᵼ (U+1D7C)ᵾ (U+1D7E)ᵿ (U+1D7F)ᶀ (U+1D80)ᶁ (U+1D81)ᶂ (U+1D82)ᶃ (U+1D83)ᶄ (U+1D84)ᶅ (U+1D85)ᶆ (U+1D86)ᶇ (U+1D87)ᶈ (U+1D88)ᶉ (U+1D89)ᶊ (U+1D8A)ᶋ (U+1D8B)ᶌ (U+1D8C)ᶍ (U+1D8D)ᶏ (U+1D8F)ᶐ (U+1D90)ᶑ (U+1D91)ᶒ (U+1D92)ᶓ (U+1D93)ᶔ (U+1D94)ᶕ (U+1D95)ᶖ (U+1D96)ᶗ (U+1D97)ᶘ (U+1D98)ᶙ (U+1D99)ᶚ (U+1D9A)ẜ (U+1E9C)ẝ (U+1E9D)ẟ (U+1E9F)ℊ (U+210A)ℎ (U+210E)ℏ (U+210F)ℓ (U+2113)ℯ (U+212F)ℴ (U+2134)ℹ (U+2139)ℼ (U+213C)ℽ (U+213D)ⅆ (U+2146)ⅇ (U+2147)ⅈ (U+2148)ⅉ (U+2149)ⱱ (U+2C71)ⱴ (U+2C74)ⱷ (U+2C77)ⱸ (U+2C78)ⱹ (U+2C79)ⱺ (U+2C7A)ⱻ (U+2C7B)ⳤ (U+2CE4)ꜰ (U+A730)ꜱ (U+A731)ꝱ (U+A771)ꝲ (U+A772)ꝳ (U+A773)ꝴ (U+A774)ꝵ (U+A775)ꝶ (U+A776)ꝷ (U+A777)ꝸ (U+A778)ꞎ (U+A78E)ꞕ (U+A795)ꞯ (U+A7AF)ꟍ (U+A7CD)꟏ (U+A7CF)ꟓ (U+A7D3)ꟕ (U+A7D5)ꟛ (U+A7DB)ꟺ (U+A7FA)ꬰ (U+AB30)ꬱ (U+AB31)ꬲ (U+AB32)ꬳ (U+AB33)ꬴ (U+AB34)ꬵ (U+AB35)ꬶ (U+AB36)ꬷ (U+AB37)ꬸ (U+AB38)ꬹ (U+AB39)ꬺ (U+AB3A)ꬻ (U+AB3B)ꬼ (U+AB3C)ꬽ (U+AB3D)ꬾ (U+AB3E)ꬿ (U+AB3F)ꭀ (U+AB40)ꭁ (U+AB41)ꭂ (U+AB42)ꭃ (U+AB43)ꭄ (U+AB44)ꭅ (U+AB45)ꭆ (U+AB46)ꭇ (U+AB47)ꭈ (U+AB48)ꭉ (U+AB49)ꭊ (U+AB4A)ꭋ (U+AB4B)ꭌ (U+AB4C)ꭍ (U+AB4D)ꭎ (U+AB4E)ꭏ (U+AB4F)ꭐ (U+AB50)ꭑ (U+AB51)ꭒ (U+AB52)ꭔ (U+AB54)ꭕ (U+AB55)ꭖ (U+AB56)ꭗ (U+AB57)ꭘ (U+AB58)ꭙ (U+AB59)ꭚ (U+AB5A)ꭠ (U+AB60)ꭡ (U+AB61)ꭢ (U+AB62)ꭣ (U+AB63)ꭤ (U+AB64)ꭥ (U+AB65)ꭦ (U+AB66)ꭧ (U+AB67)ꭨ (U+AB68)𐵰 (U+10D70)𐵱 (U+10D71)𐵲 (U+10D72)𐵳 (U+10D73)𐵴 (U+10D74)𐵵 (U+10D75)𐵶 (U+10D76)𐵷 (U+10D77)𐵸 (U+10D78)𐵹 (U+10D79)𐵺 (U+10D7A)𐵻 (U+10D7B)𐵼 (U+10D7C)𐵽 (U+10D7D)𐵾 (U+10D7E)𐵿 (U+10D7F)𐶀 (U+10D80)𐶁 (U+10D81)𐶂 (U+10D82)𐶃 (U+10D83)𐶄 (U+10D84)𐶅 (U+10D85)𖺻 (U+16EBB)𖺼 (U+16EBC)𖺽 (U+16EBD)𖺾 (U+16EBE)𖺿 (U+16EBF)𖻀 (U+16EC0)𖻁 (U+16EC1)𖻂 (U+16EC2)𖻃 (U+16EC3)𖻄 (U+16EC4)𖻅 (U+16EC5)𖻆 (U+16EC6)𖻇 (U+16EC7)𖻈 (U+16EC8)𖻉 (U+16EC9)𖻊 (U+16ECA)𖻋 (U+16ECB)𖻌 (U+16ECC)𖻍 (U+16ECD)𖻎 (U+16ECE)𖻏 (U+16ECF)𖻐 (U+16ED0)𖻑 (U+16ED1)𖻒 (U+16ED2)𖻓 (U+16ED3)𝐚 (U+1D41A)𝐛 (U+1D41B)𝐜 (U+1D41C)𝐝 (U+1D41D)𝐞 (U+1D41E)𝐟 (U+1D41F)𝐠 (U+1D420)𝐡 (U+1D421)𝐢 (U+1D422)𝐣 (U+1D423)𝐤 (U+1D424)𝐥 (U+1D425)𝐦 (U+1D426)𝐧 (U+1D427)𝐨 (U+1D428)𝐩 (U+1D429)𝐪 (U+1D42A)𝐫 (U+1D42B)𝐬 (U+1D42C)𝐭 (U+1D42D)𝐮 (U+1D42E)𝐯 (U+1D42F)𝐰 (U+1D430)𝐱 (U+1D431)𝐲 (U+1D432)𝐳 (U+1D433)𝑎 (U+1D44E)𝑏 (U+1D44F)𝑐 (U+1D450)𝑑 (U+1D451)𝑒 (U+1D452)𝑓 (U+1D453)𝑔 (U+1D454)𝑖 (U+1D456)𝑗 (U+1D457)𝑘 (U+1D458)𝑙 (U+1D459)𝑚 (U+1D45A)𝑛 (U+1D45B)𝑜 (U+1D45C)𝑝 (U+1D45D)𝑞 (U+1D45E)𝑟 (U+1D45F)𝑠 (U+1D460)𝑡 (U+1D461)𝑢 (U+1D462)𝑣 (U+1D463)𝑤 (U+1D464)𝑥 (U+1D465)𝑦 (U+1D466)𝑧 (U+1D467)𝒂 (U+1D482)𝒃 (U+1D483)𝒄 (U+1D484)𝒅 (U+1D485)𝒆 (U+1D486)𝒇 (U+1D487)𝒈 (U+1D488)𝒉 (U+1D489)𝒊 (U+1D48A)𝒋 (U+1D48B)𝒌 (U+1D48C)𝒍 (U+1D48D)𝒎 (U+1D48E)𝒏 (U+1D48F)𝒐 (U+1D490)𝒑 (U+1D491)𝒒 (U+1D492)𝒓 (U+1D493)𝒔 (U+1D494)𝒕 (U+1D495)𝒖 (U+1D496)𝒗 (U+1D497)𝒘 (U+1D498)𝒙 (U+1D499)𝒚 (U+1D49A)𝒛 (U+1D49B)𝒶 (U+1D4B6)𝒷 (U+1D4B7)𝒸 (U+1D4B8)𝒹 (U+1D4B9)𝒻 (U+1D4BB)𝒽 (U+1D4BD)𝒾 (U+1D4BE)𝒿 (U+1D4BF)𝓀 (U+1D4C0)𝓁 (U+1D4C1)𝓂 (U+1D4C2)𝓃 (U+1D4C3)𝓅 (U+1D4C5)𝓆 (U+1D4C6)𝓇 (U+1D4C7)𝓈 (U+1D4C8)𝓉 (U+1D4C9)𝓊 (U+1D4CA)𝓋 (U+1D4CB)𝓌 (U+1D4CC)𝓍 (U+1D4CD)𝓎 (U+1D4CE)𝓏 (U+1D4CF)𝓪 (U+1D4EA)𝓫 (U+1D4EB)𝓬 (U+1D4EC)𝓭 (U+1D4ED)𝓮 (U+1D4EE)𝓯 (U+1D4EF)𝓰 (U+1D4F0)𝓱 (U+1D4F1)𝓲 (U+1D4F2)𝓳 (U+1D4F3)𝓴 (U+1D4F4)𝓵 (U+1D4F5)𝓶 (U+1D4F6)𝓷 (U+1D4F7)𝓸 (U+1D4F8)𝓹 (U+1D4F9)𝓺 (U+1D4FA)𝓻 (U+1D4FB)𝓼 (U+1D4FC)𝓽 (U+1D4FD)𝓾 (U+1D4FE)𝓿 (U+1D4FF)𝔀 (U+1D500)𝔁 (U+1D501)𝔂 (U+1D502)𝔃 (U+1D503)𝔞 (U+1D51E)𝔟 (U+1D51F)𝔠 (U+1D520)𝔡 (U+1D521)𝔢 (U+1D522)𝔣 (U+1D523)𝔤 (U+1D524)𝔥 (U+1D525)𝔦 (U+1D526)𝔧 (U+1D527)𝔨 (U+1D528)𝔩 (U+1D529)𝔪 (U+1D52A)𝔫 (U+1D52B)𝔬 (U+1D52C)𝔭 (U+1D52D)𝔮 (U+1D52E)𝔯 (U+1D52F)𝔰 (U+1D530)𝔱 (U+1D531)𝔲 (U+1D532)𝔳 (U+1D533)𝔴 (U+1D534)𝔵 (U+1D535)𝔶 (U+1D536)𝔷 (U+1D537)𝕒 (U+1D552)𝕓 (U+1D553)𝕔 (U+1D554)𝕕 (U+1D555)𝕖 (U+1D556)𝕗 (U+1D557)𝕘 (U+1D558)𝕙 (U+1D559)𝕚 (U+1D55A)𝕛 (U+1D55B)𝕜 (U+1D55C)𝕝 (U+1D55D)𝕞 (U+1D55E)𝕟 (U+1D55F)𝕠 (U+1D560)𝕡 (U+1D561)𝕢 (U+1D562)𝕣 (U+1D563)𝕤 (U+1D564)𝕥 (U+1D565)𝕦 (U+1D566)𝕧 (U+1D567)𝕨 (U+1D568)𝕩 (U+1D569)𝕪 (U+1D56A)𝕫 (U+1D56B)𝖆 (U+1D586)𝖇 (U+1D587)𝖈 (U+1D588)𝖉 (U+1D589)𝖊 (U+1D58A)𝖋 (U+1D58B)𝖌 (U+1D58C)𝖍 (U+1D58D)𝖎 (U+1D58E)𝖏 (U+1D58F)𝖐 (U+1D590)𝖑 (U+1D591)𝖒 (U+1D592)𝖓 (U+1D593)𝖔 (U+1D594)𝖕 (U+1D595)𝖖 (U+1D596)𝖗 (U+1D597)𝖘 (U+1D598)𝖙 (U+1D599)𝖚 (U+1D59A)𝖛 (U+1D59B)𝖜 (U+1D59C)𝖝 (U+1D59D)𝖞 (U+1D59E)𝖟 (U+1D59F)𝖺 (U+1D5BA)𝖻 (U+1D5BB)𝖼 (U+1D5BC)𝖽 (U+1D5BD)𝖾 (U+1D5BE)𝖿 (U+1D5BF)𝗀 (U+1D5C0)𝗁 (U+1D5C1)𝗂 (U+1D5C2)𝗃 (U+1D5C3)𝗄 (U+1D5C4)𝗅 (U+1D5C5)𝗆 (U+1D5C6)𝗇 (U+1D5C7)𝗈 (U+1D5C8)𝗉 (U+1D5C9)𝗊 (U+1D5CA)𝗋 (U+1D5CB)𝗌 (U+1D5CC)𝗍 (U+1D5CD)𝗎 (U+1D5CE)𝗏 (U+1D5CF)𝗐 (U+1D5D0)𝗑 (U+1D5D1)𝗒 (U+1D5D2)𝗓 (U+1D5D3)𝗮 (U+1D5EE)𝗯 (U+1D5EF)𝗰 (U+1D5F0)𝗱 (U+1D5F1)𝗲 (U+1D5F2)𝗳 (U+1D5F3)𝗴 (U+1D5F4)𝗵 (U+1D5F5)𝗶 (U+1D5F6)𝗷 (U+1D5F7)𝗸 (U+1D5F8)𝗹 (U+1D5F9)𝗺 (U+1D5FA)𝗻 (U+1D5FB)𝗼 (U+1D5FC)𝗽 (U+1D5FD)𝗾 (U+1D5FE)𝗿 (U+1D5FF)𝘀 (U+1D600)𝘁 (U+1D601)𝘂 (U+1D602)𝘃 (U+1D603)𝘄 (U+1D604)𝘅 (U+1D605)𝘆 (U+1D606)𝘇 (U+1D607)𝘢 (U+1D622)𝘣 (U+1D623)𝘤 (U+1D624)𝘥 (U+1D625)𝘦 (U+1D626)𝘧 (U+1D627)𝘨 (U+1D628)𝘩 (U+1D629)𝘪 (U+1D62A)𝘫 (U+1D62B)𝘬 (U+1D62C)𝘭 (U+1D62D)𝘮 (U+1D62E)𝘯 (U+1D62F)𝘰 (U+1D630)𝘱 (U+1D631)𝘲 (U+1D632)𝘳 (U+1D633)𝘴 (U+1D634)𝘵 (U+1D635)𝘶 (U+1D636)𝘷 (U+1D637)𝘸 (U+1D638)𝘹 (U+1D639)𝘺 (U+1D63A)𝘻 (U+1D63B)𝙖 (U+1D656)𝙗 (U+1D657)𝙘 (U+1D658)𝙙 (U+1D659)𝙚 (U+1D65A)𝙛 (U+1D65B)𝙜 (U+1D65C)𝙝 (U+1D65D)𝙞 (U+1D65E)𝙟 (U+1D65F)𝙠 (U+1D660)𝙡 (U+1D661)𝙢 (U+1D662)𝙣 (U+1D663)𝙤 (U+1D664)𝙥 (U+1D665)𝙦 (U+1D666)𝙧 (U+1D667)𝙨 (U+1D668)𝙩 (U+1D669)𝙪 (U+1D66A)𝙫 (U+1D66B)𝙬 (U+1D66C)𝙭 (U+1D66D)𝙮 (U+1D66E)𝙯 (U+1D66F)𝚊 (U+1D68A)𝚋 (U+1D68B)𝚌 (U+1D68C)𝚍 (U+1D68D)𝚎 (U+1D68E)𝚏 (U+1D68F)𝚐 (U+1D690)𝚑 (U+1D691)𝚒 (U+1D692)𝚓 (U+1D693)𝚔 (U+1D694)𝚕 (U+1D695)𝚖 (U+1D696)𝚗 (U+1D697)𝚘 (U+1D698)𝚙 (U+1D699)𝚚 (U+1D69A)𝚛 (U+1D69B)𝚜 (U+1D69C)𝚝 (U+1D69D)𝚞 (U+1D69E)𝚟 (U+1D69F)𝚠 (U+1D6A0)𝚡 (U+1D6A1)𝚢 (U+1D6A2)𝚣 (U+1D6A3)𝚤 (U+1D6A4)𝚥 (U+1D6A5)𝛂 (U+1D6C2)𝛃 (U+1D6C3)𝛄 (U+1D6C4)𝛅 (U+1D6C5)𝛆 (U+1D6C6)𝛇 (U+1D6C7)𝛈 (U+1D6C8)𝛉 (U+1D6C9)𝛊 (U+1D6CA)𝛋 (U+1D6CB)𝛌 (U+1D6CC)𝛍 (U+1D6CD)𝛎 (U+1D6CE)𝛏 (U+1D6CF)𝛐 (U+1D6D0)𝛑 (U+1D6D1)𝛒 (U+1D6D2)𝛓 (U+1D6D3)𝛔 (U+1D6D4)𝛕 (U+1D6D5)𝛖 (U+1D6D6)𝛗 (U+1D6D7)𝛘 (U+1D6D8)𝛙 (U+1D6D9)𝛚 (U+1D6DA)𝛜 (U+1D6DC)𝛝 (U+1D6DD)𝛞 (U+1D6DE)𝛟 (U+1D6DF)𝛠 (U+1D6E0)𝛡 (U+1D6E1)𝛼 (U+1D6FC)𝛽 (U+1D6FD)𝛾 (U+1D6FE)𝛿 (U+1D6FF)𝜀 (U+1D700)𝜁 (U+1D701)𝜂 (U+1D702)𝜃 (U+1D703)𝜄 (U+1D704)𝜅 (U+1D705)𝜆 (U+1D706)𝜇 (U+1D707)𝜈 (U+1D708)𝜉 (U+1D709)𝜊 (U+1D70A)𝜋 (U+1D70B)𝜌 (U+1D70C)𝜍 (U+1D70D)𝜎 (U+1D70E)𝜏 (U+1D70F)𝜐 (U+1D710)𝜑 (U+1D711)𝜒 (U+1D712)𝜓 (U+1D713)𝜔 (U+1D714)𝜖 (U+1D716)𝜗 (U+1D717)𝜘 (U+1D718)𝜙 (U+1D719)𝜚 (U+1D71A)𝜛 (U+1D71B)𝜶 (U+1D736)𝜷 (U+1D737)𝜸 (U+1D738)𝜹 (U+1D739)𝜺 (U+1D73A)𝜻 (U+1D73B)𝜼 (U+1D73C)𝜽 (U+1D73D)𝜾 (U+1D73E)𝜿 (U+1D73F)𝝀 (U+1D740)𝝁 (U+1D741)𝝂 (U+1D742)𝝃 (U+1D743)𝝄 (U+1D744)𝝅 (U+1D745)𝝆 (U+1D746)𝝇 (U+1D747)𝝈 (U+1D748)𝝉 (U+1D749)𝝊 (U+1D74A)𝝋 (U+1D74B)𝝌 (U+1D74C)𝝍 (U+1D74D)𝝎 (U+1D74E)𝝐 (U+1D750)𝝑 (U+1D751)𝝒 (U+1D752)𝝓 (U+1D753)𝝔 (U+1D754)𝝕 (U+1D755)𝝰 (U+1D770)𝝱 (U+1D771)𝝲 (U+1D772)𝝳 (U+1D773)𝝴 (U+1D774)𝝵 (U+1D775)𝝶 (U+1D776)𝝷 (U+1D777)𝝸 (U+1D778)𝝹 (U+1D779)𝝺 (U+1D77A)𝝻 (U+1D77B)𝝼 (U+1D77C)𝝽 (U+1D77D)𝝾 (U+1D77E)𝝿 (U+1D77F)𝞀 (U+1D780)𝞁 (U+1D781)𝞂 (U+1D782)𝞃 (U+1D783)𝞄 (U+1D784)𝞅 (U+1D785)𝞆 (U+1D786)𝞇 (U+1D787)𝞈 (U+1D788)𝞊 (U+1D78A)𝞋 (U+1D78B)𝞌 (U+1D78C)𝞍 (U+1D78D)𝞎 (U+1D78E)𝞏 (U+1D78F)𝞪 (U+1D7AA)𝞫 (U+1D7AB)𝞬 (U+1D7AC)𝞭 (U+1D7AD)𝞮 (U+1D7AE)𝞯 (U+1D7AF)𝞰 (U+1D7B0)𝞱 (U+1D7B1)𝞲 (U+1D7B2)𝞳 (U+1D7B3)𝞴 (U+1D7B4)𝞵 (U+1D7B5)𝞶 (U+1D7B6)𝞷 (U+1D7B7)𝞸 (U+1D7B8)𝞹 (U+1D7B9)𝞺 (U+1D7BA)𝞻 (U+1D7BB)𝞼 (U+1D7BC)𝞽 (U+1D7BD)𝞾 (U+1D7BE)𝞿 (U+1D7BF)𝟀 (U+1D7C0)𝟁 (U+1D7C1)𝟂 (U+1D7C2)𝟄 (U+1D7C4)𝟅 (U+1D7C5)𝟆 (U+1D7C6)𝟇 (U+1D7C7)𝟈 (U+1D7C8)𝟉 (U+1D7C9)𝟋 (U+1D7CB)𝼀 (U+1DF00)𝼁 (U+1DF01)𝼂 (U+1DF02)𝼃 (U+1DF03)𝼄 (U+1DF04)𝼅 (U+1DF05)𝼆 (U+1DF06)𝼇 (U+1DF07)𝼈 (U+1DF08)𝼉 (U+1DF09)𝼋 (U+1DF0B)𝼌 (U+1DF0C)𝼍 (U+1DF0D)𝼎 (U+1DF0E)𝼏 (U+1DF0F)𝼐 (U+1DF10)𝼑 (U+1DF11)𝼒 (U+1DF12)𝼓 (U+1DF13)𝼔 (U+1DF14)𝼕 (U+1DF15)𝼖 (U+1DF16)𝼗 (U+1DF17)𝼘 (U+1DF18)𝼙 (U+1DF19)𝼚 (U+1DF1A)𝼛 (U+1DF1B)𝼜 (U+1DF1C)𝼝 (U+1DF1D)𝼞 (U+1DF1E)𝼥 (U+1DF25)𝼦 (U+1DF26)𝼧 (U+1DF27)𝼨 (U+1DF28)𝼩 (U+1DF29)𝼪 (U+1DF2A)

This means there are cased letters that are case-mapping invariant. Always-upper characters include:

• Greek and Coptic
  • Variants of GREEK UPSILON: U+03D2 – U+03D4
• Letterlike Symbols
  • EULER CONSTANT (U+2107)
  • DOUBLE-STRUCK (ITALIC) CAPITAL {C,H,N,P,Q,R,Z,GAMMA,PI,D}: U+2102, U+210D, U+2115, U+2119, U+211A, U+211D, U+2124, U+213E, U+213F, U+2145
  • SCRIPT CAPITAL {H,I,L,R,B,E,F,M}: U+210B, U+2110, U+2112, U+211B, U+212C, U+2130, U+2131, U+2133
  • BLACK-LETTER CAPITAL {H,I,R,Z,C}: U+210C, U+2111, U+211C, U+2128, U+212D
• Mathematical Alphanumeric Symbols
  • MATHEMATICAL {BOLD,ITALIC,BOLD ITALIC,SCRIPT,BOLD SCRIPT,FRAKTUR,DOUBLE-STRUCK,BOLD FRAKTUR,SANS-SERIF,SANS-SERIF BOLD,SANS-SERIF ITALIC,SANS-SERIF BOLD ITALIC,MONOSPACE} CAPITAL Latin alphabet: U+1D400 – U+1D419, U+1D434 – U+1D44D, U+1D468 – U+1D481, U+1D49C – U+1D4B5, U+1D4D0 – U+1D4E9, U+1D504 – U+1D51C, U+1D538 – U+1D550, U+1D56C – U+1D585, U+1D5A0 – U+1D5B9, U+1D5D4 – U+1D5ED, U+1D608 – U+1D621, U+1D63C – U+1D655, U+1D670 – U+1D689
  • MATHEMATICAL {BOLD,ITALIC,BOLD ITALIC,SANS-SERIF BOLD,SANS-SERIF ITALIC} CAPITAL Greek alphabet: U+1D6A8 – U+1D6C0, U+1D6E2 – U+1D6FA, U+1D71C – U+1D734, U+1D756 – U+1D76E, U+1D790 – U+1D7A8
  • MATHEMATICAL BOLD CAPITAL DIGAMMA (U+1D7CA)

Always-lower characters include:

• Latin Extended-A
  • LATIN SMALL LETTER KRA (U+0138)
• Latin Extended-B
  • LATIN SMALL LETTER TURNED DELTA (U+018D)
  • LATIN SMALL LETTER LAMBDA WITH STROKE (U+019B)
  • LATIN LETTER REVERSED ESH LOOP (U+01AA)
  • LATIN SMALL LETTER T WITH PALATAL HOOK (U+01AB)
  • LATIN SMALL LETTER EZH WITH TAIL (U+01BA)
  • LATIN LETTER INVERTED GLOTTAL STOP WITH STROKE (U+01BE)
  • LATIN SMALL LETTER {D,L,N,T} WITH CURL: U+0221, U+0234 – U+0236
  • LATIN SMALL LETTER DOTLESS J (U+0237)
  • LATIN SMALL LETTER {DB,QP} DIGRAPH: U+0238, U+0239
• IPA Extensions
  • U+0250 – U+02AF, except 28 of them
• Greek and Coptic
  • GREEK RHO WITH STROKE SYMBOL (U+03FC)
• Armenian
  • ARMENIAN SMALL LETTER TURNED AYB (U+0560)
  • ARMENIAN SMALL LETTER YI WITH STROKE (U+0589)
• Phonetic Extensions
  • Latin letters, Greek letters, and Cyrillic letter: U+1D00 – U+1D2B
  • Latin letter for American lexicography, Latin letters with middle tilde: U+1D6B – U+1D76
  • LATIN SMALL LETTER TURNED G (U+1D77)
  • Other phonetic symbols, except LATIN SMALL LETTER INSULAR G: U+1D7A – U+1D7F
• Phonetic Extensions Supplement
  • Latin letters with palatal hook, except LATIN SMALL LETTER Z WITH PALATAL HOOK: U+1D80 – U+1D8D
  • Latin letters with retroflex hook: U+1D8F – U+1D9A
• Latin Extended Additional
  • Mediavalist additions: U+1E9C, U+1E9D, U+1E9F
• ...

To make our discussions more meaningful, we will limit our future discussions to $U$ and $L$ instead of $U_{ext}$ and $L_{ext}$, so that all sets in question are subsets of $\mathbb{UC}$. Upper(Lower) case letters that are case-mapping variant must be lower(upper)case variant, because we already showed that they are upper(lower)case invariant.

Does case-mapping variance imply casedness?

No. $U_{ext}$ and $L_{ext}$ are not partitions of $\mathbb{UC}$: there are characters that are uncased, but are case-mapping variant:

• $\mathbb{UC}\setminus(U_{ext}\cup L_{ext})$ =
  Character set (116)

  Dž (U+01C5)Lj (U+01C8)Nj (U+01CB)Dz (U+01F2)ͅ (U+0345)ᾈ (U+1F88)ᾉ (U+1F89)ᾊ (U+1F8A)ᾋ (U+1F8B)ᾌ (U+1F8C)ᾍ (U+1F8D)ᾎ (U+1F8E)ᾏ (U+1F8F)ᾘ (U+1F98)ᾙ (U+1F99)ᾚ (U+1F9A)ᾛ (U+1F9B)ᾜ (U+1F9C)ᾝ (U+1F9D)ᾞ (U+1F9E)ᾟ (U+1F9F)ᾨ (U+1FA8)ᾩ (U+1FA9)ᾪ (U+1FAA)ᾫ (U+1FAB)ᾬ (U+1FAC)ᾭ (U+1FAD)ᾮ (U+1FAE)ᾯ (U+1FAF)ᾼ (U+1FBC)ῌ (U+1FCC)ῼ (U+1FFC)Ⅰ (U+2160)Ⅱ (U+2161)Ⅲ (U+2162)Ⅳ (U+2163)Ⅴ (U+2164)Ⅵ (U+2165)Ⅶ (U+2166)Ⅷ (U+2167)Ⅸ (U+2168)Ⅹ (U+2169)Ⅺ (U+216A)Ⅻ (U+216B)Ⅼ (U+216C)Ⅽ (U+216D)Ⅾ (U+216E)Ⅿ (U+216F)ⅰ (U+2170)ⅱ (U+2171)ⅲ (U+2172)ⅳ (U+2173)ⅴ (U+2174)ⅵ (U+2175)ⅶ (U+2176)ⅷ (U+2177)ⅸ (U+2178)ⅹ (U+2179)ⅺ (U+217A)ⅻ (U+217B)ⅼ (U+217C)ⅽ (U+217D)ⅾ (U+217E)ⅿ (U+217F)Ⓐ (U+24B6)Ⓑ (U+24B7)Ⓒ (U+24B8)Ⓓ (U+24B9)Ⓔ (U+24BA)Ⓕ (U+24BB)Ⓖ (U+24BC)Ⓗ (U+24BD)Ⓘ (U+24BE)Ⓙ (U+24BF)Ⓚ (U+24C0)Ⓛ (U+24C1)Ⓜ (U+24C2)Ⓝ (U+24C3)Ⓞ (U+24C4)Ⓟ (U+24C5)Ⓠ (U+24C6)Ⓡ (U+24C7)Ⓢ (U+24C8)Ⓣ (U+24C9)Ⓤ (U+24CA)Ⓥ (U+24CB)Ⓦ (U+24CC)Ⓧ (U+24CD)Ⓨ (U+24CE)Ⓩ (U+24CF)ⓐ (U+24D0)ⓑ (U+24D1)ⓒ (U+24D2)ⓓ (U+24D3)ⓔ (U+24D4)ⓕ (U+24D5)ⓖ (U+24D6)ⓗ (U+24D7)ⓘ (U+24D8)ⓙ (U+24D9)ⓚ (U+24DA)ⓛ (U+24DB)ⓜ (U+24DC)ⓝ (U+24DD)ⓞ (U+24DE)ⓟ (U+24DF)ⓠ (U+24E0)ⓡ (U+24E1)ⓢ (U+24E2)ⓣ (U+24E3)ⓤ (U+24E4)ⓥ (U+24E5)ⓦ (U+24E6)ⓧ (U+24E7)ⓨ (U+24E8)ⓩ (U+24E9)

They include $Gr_U$, $La$, and also:

• Number Forms
  • Uppercase roman numerals ($Ro_U$): U+2160 – U+216F
  • Small roman numerals ($Ro_L$): U+2170 – U+217F
• Enclosed Alphanumerics
  • Circled Latin letters ($Ci_U$): U+24B6 – U+24CF
  • Circled small Latin letters ($Ci_L$): U+24D0 – U+24E9
• Combining Diacritical Marks
  • COMBINING GREEK YPOGEGRAMMENI (U+0345) (Previously mentioned)

Are uppercase variance and lowercase variance mutually exclusive?

No. There are characters that are both uppercase variant and lowercase variant:

• $\{ c\in\mathbb{UC}\mid c\neq \mathtt{toUpperCase}(c) \neq \mathtt{toLowerCase}(c) \}$ = $La\cup Gr_U$ =
  Character set (31)

  • Dž (U+01C5) → DŽ (U+01C4), dž (U+01C6)
  • Lj (U+01C8) → LJ (U+01C7), lj (U+01C9)
  • Nj (U+01CB) → NJ (U+01CA), nj (U+01CC)
  • Dz (U+01F2) → DZ (U+01F1), dz (U+01F3)
  • ᾈ (U+1F88) → ἈΙ (U+1F08 U+0399), ᾀ (U+1F80)
  • ᾉ (U+1F89) → ἉΙ (U+1F09 U+0399), ᾁ (U+1F81)
  • ᾊ (U+1F8A) → ἊΙ (U+1F0A U+0399), ᾂ (U+1F82)
  • ᾋ (U+1F8B) → ἋΙ (U+1F0B U+0399), ᾃ (U+1F83)
  • ᾌ (U+1F8C) → ἌΙ (U+1F0C U+0399), ᾄ (U+1F84)
  • ᾍ (U+1F8D) → ἍΙ (U+1F0D U+0399), ᾅ (U+1F85)
  • ᾎ (U+1F8E) → ἎΙ (U+1F0E U+0399), ᾆ (U+1F86)
  • ᾏ (U+1F8F) → ἏΙ (U+1F0F U+0399), ᾇ (U+1F87)
  • ᾘ (U+1F98) → ἨΙ (U+1F28 U+0399), ᾐ (U+1F90)
  • ᾙ (U+1F99) → ἩΙ (U+1F29 U+0399), ᾑ (U+1F91)
  • ᾚ (U+1F9A) → ἪΙ (U+1F2A U+0399), ᾒ (U+1F92)
  • ᾛ (U+1F9B) → ἫΙ (U+1F2B U+0399), ᾓ (U+1F93)
  • ᾜ (U+1F9C) → ἬΙ (U+1F2C U+0399), ᾔ (U+1F94)
  • ᾝ (U+1F9D) → ἭΙ (U+1F2D U+0399), ᾕ (U+1F95)
  • ᾞ (U+1F9E) → ἮΙ (U+1F2E U+0399), ᾖ (U+1F96)
  • ᾟ (U+1F9F) → ἯΙ (U+1F2F U+0399), ᾗ (U+1F97)
  • ᾨ (U+1FA8) → ὨΙ (U+1F68 U+0399), ᾠ (U+1FA0)
  • ᾩ (U+1FA9) → ὩΙ (U+1F69 U+0399), ᾡ (U+1FA1)
  • ᾪ (U+1FAA) → ὪΙ (U+1F6A U+0399), ᾢ (U+1FA2)
  • ᾫ (U+1FAB) → ὫΙ (U+1F6B U+0399), ᾣ (U+1FA3)
  • ᾬ (U+1FAC) → ὬΙ (U+1F6C U+0399), ᾤ (U+1FA4)
  • ᾭ (U+1FAD) → ὭΙ (U+1F6D U+0399), ᾥ (U+1FA5)
  • ᾮ (U+1FAE) → ὮΙ (U+1F6E U+0399), ᾦ (U+1FA6)
  • ᾯ (U+1FAF) → ὯΙ (U+1F6F U+0399), ᾧ (U+1FA7)
  • ᾼ (U+1FBC) → ΑΙ (U+0391 U+0399), ᾳ (U+1FB3)
  • ῌ (U+1FCC) → ΗΙ (U+0397 U+0399), ῃ (U+1FC3)
  • ῼ (U+1FFC) → ΩΙ (U+03A9 U+0399), ῳ (U+1FF3)

In addition, as shown before, these are also characters that cannot be produced by toUpperCase() or toLowerCase() with any input, including themselves.

Is Lower(Upper)Case_Letter always mapped to Upper(Lower)Case_Letter by toUpper(Lower)Case?

We already mentioned that certain upper-/lower-case letters are mapping invariant. Furthermore, there are plenty of characters in $M_L\cup M_U\cup N_L\cup N_U$ that are cased. Dropping those, the answer is yes. If the input is a Lowercase_Letter, the output of toUpperCase() is always an Uppercase_Letter. If the input is an Uppercase_Letter, the output of toLowerCase() is always a Lowercase_Letter.

• $\mathtt{toUpperCase}(L\setminus M_L\setminus N_L) \setminus U$ = ∅
• $\mathtt{toLowerCase}(U\setminus M_U\setminus N_U) \setminus L$ = ∅

Does toUpper(Lower)Case always produce Upper(Lower)case_Letter? Can it produce Lower(Upper)case_Letter?

(Again, disregarding multi-code-point characters) No and no (but yes, if you count case-mapping invariant but cased characters). $U$ and $L$ are proper subsets of $\mathcal{R}_U$ and $\mathcal{R}_L$, respectively:

• $U\setminus \mathcal{R}_U$ = ∅
• $\mathcal{R}_U\setminus U\setminus M_L'\setminus N_L'$ = $Ro_U\cup Ci_U$ =
  Character set (42)

  Ⅰ (U+2160)Ⅱ (U+2161)Ⅲ (U+2162)Ⅳ (U+2163)Ⅴ (U+2164)Ⅵ (U+2165)Ⅶ (U+2166)Ⅷ (U+2167)Ⅸ (U+2168)Ⅹ (U+2169)Ⅺ (U+216A)Ⅻ (U+216B)Ⅼ (U+216C)Ⅽ (U+216D)Ⅾ (U+216E)Ⅿ (U+216F)Ⓐ (U+24B6)Ⓑ (U+24B7)Ⓒ (U+24B8)Ⓓ (U+24B9)Ⓔ (U+24BA)Ⓕ (U+24BB)Ⓖ (U+24BC)Ⓗ (U+24BD)Ⓘ (U+24BE)Ⓙ (U+24BF)Ⓚ (U+24C0)Ⓛ (U+24C1)Ⓜ (U+24C2)Ⓝ (U+24C3)Ⓞ (U+24C4)Ⓟ (U+24C5)Ⓠ (U+24C6)Ⓡ (U+24C7)Ⓢ (U+24C8)Ⓣ (U+24C9)Ⓤ (U+24CA)Ⓥ (U+24CB)Ⓦ (U+24CC)Ⓧ (U+24CD)Ⓨ (U+24CE)Ⓩ (U+24CF)
• $L\setminus \mathcal{R}_L$ = ∅
• $\mathcal{R}_L\setminus L\setminus M_U'\setminus N_U'$ = $Ro_L\cup Ci_L\cup\{\text{U+0345}\}$ =
  Character set (43)

  ͅ (U+0345)ⅰ (U+2170)ⅱ (U+2171)ⅲ (U+2172)ⅳ (U+2173)ⅴ (U+2174)ⅵ (U+2175)ⅶ (U+2176)ⅷ (U+2177)ⅸ (U+2178)ⅹ (U+2179)ⅺ (U+217A)ⅻ (U+217B)ⅼ (U+217C)ⅽ (U+217D)ⅾ (U+217E)ⅿ (U+217F)ⓐ (U+24D0)ⓑ (U+24D1)ⓒ (U+24D2)ⓓ (U+24D3)ⓔ (U+24D4)ⓕ (U+24D5)ⓖ (U+24D6)ⓗ (U+24D7)ⓘ (U+24D8)ⓙ (U+24D9)ⓚ (U+24DA)ⓛ (U+24DB)ⓜ (U+24DC)ⓝ (U+24DD)ⓞ (U+24DE)ⓟ (U+24DF)ⓠ (U+24E0)ⓡ (U+24E1)ⓢ (U+24E2)ⓣ (U+24E3)ⓤ (U+24E4)ⓥ (U+24E5)ⓦ (U+24E6)ⓧ (U+24E7)ⓨ (U+24E8)ⓩ (U+24E9)

Uncased letters produced by to{Upper,Lower}Case are the same sets as discussed before: those characters that are uncased but case-mapping variant. To produce an uncased output, the input must be uncased too.

On the other hand, $L$ and $\mathcal{R}_U$, $U$ and $\mathcal{R}_L$ are disjoint:

• $L\cap \mathcal{R}_U$ = ∅
• $U\cap \mathcal{R}_L$ = ∅

So toUpperCase() never produces a Lowercase_Letter, and toLowerCase() never produces an Uppercase_Letter.

Can toUpper(Lower)Case produce Upper(Lower)case_Letter from uncased characters?

Yes. Uncased letters may become cased after case mapping:

• $\mathtt{toUpperCase}\left(\mathbb{UC}\setminus (U\cup L)\right) \cap U$ = $\mathtt{toUpperCase}(La\cup \{\text{U+0345}\})$ =
  Character set (5)

  DŽ (U+01C4)LJ (U+01C7)NJ (U+01CA)DZ (U+01F1)Ι (U+0399)
• $\mathtt{toLowerCase}\left(\mathbb{UC}\setminus (U\cup L)\right) \cap L$ = $\mathtt{toLowerCase}(La\cup Gr_U)$ =
  Character set (31)

  dž (U+01C6)lj (U+01C9)nj (U+01CC)dz (U+01F3)ᾀ (U+1F80)ᾁ (U+1F81)ᾂ (U+1F82)ᾃ (U+1F83)ᾄ (U+1F84)ᾅ (U+1F85)ᾆ (U+1F86)ᾇ (U+1F87)ᾐ (U+1F90)ᾑ (U+1F91)ᾒ (U+1F92)ᾓ (U+1F93)ᾔ (U+1F94)ᾕ (U+1F95)ᾖ (U+1F96)ᾗ (U+1F97)ᾠ (U+1FA0)ᾡ (U+1FA1)ᾢ (U+1FA2)ᾣ (U+1FA3)ᾤ (U+1FA4)ᾥ (U+1FA5)ᾦ (U+1FA6)ᾧ (U+1FA7)ᾳ (U+1FB3)ῃ (U+1FC3)ῳ (U+1FF3)

These characters are also the characters that are uncased but case-mapping variant.

To summarize:

Properties of characters that map to multiple characters

We now focus on these particular subsets:

• $M_L$
  • Lower case (thus lowercase invariant): $M_L\cap (\mathbb{UC}\setminus L)$ = ∅
• $M_U$
  • Upper case (thus uppercase invariant): $M_U\cap (\mathbb{UC}\setminus U)$ = ∅
• $N_L$
  • We already mentioned that $Gr_U$ is uncased, and both uppercase and lowercase variant. $N_L\setminus Gr_U$ is lower case (thus lowercase invariant): $N_L\setminus Gr_U\cap (\mathbb{UC}\setminus L)$ = ∅
• $N_U$
  • Empty set

Define $\mathbb{UC}_S = \mathbb{UC}\setminus M_U\setminus N_U\setminus M_L\setminus N_L$. So, we may characterize the domain and codomain of toUpperCase() and toLowerCase() as the following, where each piece has a disjoint domain and each except the last has a disjoint codomain:

There are other cases where multiple code points can be mapped to single code points, but they are not of our interest. We will discuss these multi-code-point characters soon.

We wonder if characters in $\mathbb{UC}_S$ always stay in $\mathbb{UC}_S$. In order to narrow the codomain of the pieces marked with (*), we want to find if there are characters $c\in \mathbb{UC}_S$ such that $\mathtt{toUpperCase}(c)\in M_U\cup M_L\cup N_U\cup N_L$, or $\mathtt{toLowerCase}(c)\in M_U\cup M_L\cup N_U\cup N_L$.

• $\{ c\in \mathbb{UC}_s\mid \mathtt{toUpperCase}(c)\in M_U\cup M_L\cup N_U\cup N_L \}$ = ∅
• $\{ c\in \mathbb{UC}_s\mid \mathtt{toLowerCase}(c)\in M_U\cup M_L\cup N_U\cup N_L \}$ =
  Character set (1)

  ẞ (U+1E9E)

There is exactly one: LATIN CAPITAL LETTER SHARP S (U+1E9E). The mappings of this character are:

• ẞ $\xrightarrow{\mathtt{toUpperCase}}$ ẞ (U+1E9E)
• ẞ $\xrightarrow{\mathtt{toLowerCase}}$ ß (U+00DF) $\xrightarrow{\mathtt{toUpperCase}}$ SS (U+0053 U+0053)

i.e. it maps to a character in $N_L$. There's no other character that maps to U+00DF:

• $\{ c\in \mathbb{UC}\mid \mathtt{toLowerCase}(c) = \text{U+00DF} \}$ =
  Character set (2)

  ß (U+00DF)ẞ (U+1E9E)

The characters that map to $\mathtt{toLowerCase}(Gr_U)$ are exactly $Gr_L$ and $Gr_U$:

• $\{ c\in \mathbb{UC}\mid \mathtt{toLowerCase}(c) \in \mathtt{toLowerCase}(Gr_U) \}$ =
  Character set (54)

  ᾀ (U+1F80)ᾁ (U+1F81)ᾂ (U+1F82)ᾃ (U+1F83)ᾄ (U+1F84)ᾅ (U+1F85)ᾆ (U+1F86)ᾇ (U+1F87)ᾈ (U+1F88)ᾉ (U+1F89)ᾊ (U+1F8A)ᾋ (U+1F8B)ᾌ (U+1F8C)ᾍ (U+1F8D)ᾎ (U+1F8E)ᾏ (U+1F8F)ᾐ (U+1F90)ᾑ (U+1F91)ᾒ (U+1F92)ᾓ (U+1F93)ᾔ (U+1F94)ᾕ (U+1F95)ᾖ (U+1F96)ᾗ (U+1F97)ᾘ (U+1F98)ᾙ (U+1F99)ᾚ (U+1F9A)ᾛ (U+1F9B)ᾜ (U+1F9C)ᾝ (U+1F9D)ᾞ (U+1F9E)ᾟ (U+1F9F)ᾠ (U+1FA0)ᾡ (U+1FA1)ᾢ (U+1FA2)ᾣ (U+1FA3)ᾤ (U+1FA4)ᾥ (U+1FA5)ᾦ (U+1FA6)ᾧ (U+1FA7)ᾨ (U+1FA8)ᾩ (U+1FA9)ᾪ (U+1FAA)ᾫ (U+1FAB)ᾬ (U+1FAC)ᾭ (U+1FAD)ᾮ (U+1FAE)ᾯ (U+1FAF)ᾳ (U+1FB3)ᾼ (U+1FBC)ῃ (U+1FC3)ῌ (U+1FCC)ῳ (U+1FF3)ῼ (U+1FFC)

Then we can refine the domain and codomain of toUpperCase() and toLowerCase() as the following, so that each piece has a disjoint domain and codomain:

Mapping graph

To study the injectivity/surjectivity of case mapping, we introduce the concept of a mapping graph, a directed graph $(\mathbb{UC}_S\cup M_L\cup M_U\cup N_L\cup N_U\cup M_L'\cup M_U'\cup N_L'\cup N_U'\cup I, E_u\cup E_l)$. The vertices are all characters that we discussed, and the edges have two colors: $(c_1, c_2)\in E_u$ iff $\mathtt{toUpperCase}(c_1) = c_2$, $(c_1, c_2)\in E_l$ iff $\mathtt{toLowerCase}(c_1) = c_2$. Therefore, we can reformulate case-mapping variance as:

1. A node is case-mapping invariant iff both of its out edges are self-loops. These are nodes in $I$.
2. A node is uppercase invariant iff its out $u$-edge is a self-loop.
3. A node is lowercase invariant iff its out $l$-edge is a self-loop.
4. A node is both uppercase and lowercase variant iff both of its out edges are not self-loops. These are the 31 characters mentioned before: $La\cup Gr_U$.

We also have the following observations:

1. Edges can be self-referential. Each node except those in $N_L'$ and $N_U'$ has exactly one out $u$-edge and exactly one out $l$-edge. $N_L'$ has a self-referential out $u$-edge and no out $l$-edge. $N_U'$ has a self-referential out $l$-edge and no out $u$-edge.
2. Each node has zero or more in edges colored either $u$ or $l$.
3. Idempotence: if a node has a non-self-referential in $u$-edge, then its out $u$-edge is self-referential. Similarly, if a node has a non-self-referential in $l$-edge, then its out $l$-edge is self-referential.
4. Complementary ranges: if a node has a non-self-referential in $u$-edge, then it has no non-self-referential in $l$-edge. Similarly, if a node has a non-self-referential in $l$-edge, then it has no non-self-referential in $u$-edge.
5. Closedness of $I$: if a node has a non-self-referential in $u$-edge, then it has no out $l$-edge or its out $l$-edge is non-self-referential. Similarly, if a node has a non-self-referential in edge colored $l$, then it has no out $u$-edge or its out $u$-edge is non-self-referential.

We already established that:

1. Nodes in $I$ have no in edges from other nodes.
2. Nodes in $La\cup Gr_U$ have no in edges from other nodes.

Therefore, ignoring these nodes, the graph is inherently bipartite:

• Each uppercase invariant node points to a lowercase invariant node with an $l$-edge and points to itself with a $u$-edge. It has zero or more in $u$-edges and no in $l$-edge.
• Each lowercase invariant node points to an uppercase invariant node with a $u$-edge and points to itself with an $l$-edge. It has zero or more in $l$-edges and no in $u$-edge.

Connected subgraphs

A connected subgraph, called a cluster, formed by $c$ is recursively defined:

1. $c$ is in the cluster.
2. If $c'$ is in the cluster, then all nodes that point to $c'$ and those that $c'$ points to are all in the cluster.

Nodes that are in the cluster formed by $c\in N_L'\cup N_U'$ do not always form cycles through some series of toUpperCase/toLowerCase transformations, because they eventually map to a node in $N_L'\cup N_U'$, which has no out $l$-edge or no out $u$-edge.

Simple mapping pairs

Define a mapping pair as a cluster of size 2. A mapping pair $(c_u, c_l)$ ($c_u\neq c_l$) satisfies:

1. $(c_u, c_l)\in E_l$ (the $l$-edge of $c_u$ points to $c_l$)
2. $(c_l, c_u)\in E_u$ (the $u$-edge of $c_l$ points to $c_u$)
3. $\nexists c\neq c_u$ such that $(c, c_u)\in E_u$ or $(c, c_l)\in E_l$ (no other node points to $c_u$ or $c_l$)

There are 1386 such pairs. Of these, 1322 are pairs of Uppercase_Letter and Lowercase_Letter, and the rest are:

Which are roman numerals $(Ro_U, Ro_L)$, circled letters $(Ci_U, Ci_L)$, and $(M_L', M_L)$, $(M_U, M_U')$.

Now, the remaining nodes are the complex cycles that neither have dead ends nor are simple mapping pairs. They are:

Which characters are the upper(lower)case form of multiple characters?

Below are all characters that is the uppercase form of multiple characters.

Below are all characters that is the lowercase form of multiple characters.