OpenLLM-France/wikimedia huggingface dataset:
# Alphabet/Spanish alphabet

The Spanish alphabet has 27 letters. Synonyms: alfabeto (from Ancient Greekἄλφα/alfa/alpha, beta...) and abecedario (from Spanish sounding: a-be-ce-da-rio).
Notes:
* Only vocals may contain an accent ( ' ). Only the letter "u" may show an accent or dieresis (ü).
* The letter "H" or "h" is silent, unless it is preceded by the letter "C" or "c" to form "Ch" (or by the letter S to form Sh).
* The combination of the letters P or p and h used in English is not used in Spanish, particularly to produce the f sound; so words like alpha (in English) are written as alfa in Spanish.
* The letter "Q" or "q" is basically accompanied by the letter "u", to be followed by a vocal, such as "e" or "i" to form the syllable que and qui.
* The letter "W" or "w", as the combination "Sh" or "sh" (and in some instances the letter "x" or "x") present flexibility to accommodate some neologisms, anglicisms, and other terms and names not necessarily original in Spanish. For instance: Washington.
* The letter "X" or "x", encompasses different sounds in pronunciation depending on the use of the word and the position of the letter in the word.
| # | Letter (Uppercase) | Letter (Lowercase) | Pronunciation (Name of the letter) |
| 1 | A | a (á) | a |
| 2 | B | b | be |
| 3 | C | c | ce |
| 4 | D | d | de |
| 5 | E | e (é) | e |
| 6 | F | f | efe |
| 7 | G | g | ge |
| 8 | H | h | atcheh |
| 9 | I | i (í) | i |
| 10 | J | j | jota |
| 11 | K | k | ka |
| 12 | L | l | ele |
| 13 | M | m | eme |
| 14 | N | n | ene |
| 15 | Ñ | ñ | eñe |
| 16 | O | o (ó) | o |
| 17 | P | p | pe |
| 18 | Q | q | coo |
| 19 | R | r | ere |
| 20 | S | s | ese |
| 21 | T | t | te |
| 22 | U | u (ú, ü) | u |
| 23 | V | v | uve |
| 24 | W | w | doble uve |
| 25 | X | x | equis |
| 26 | Y | y | ye, i griega |
| 27 | Z | z | zeta |
* Wikipedia: Spanish alphabet
* Portal: Spanish
# Boolf prop/3-ary/serration 1 (quarter sharpness)

The matrices marked with Я show permutations. That for (0, 0, 0, 0) shows Walsh permutation (2, 3, 8, 12).
Number of blocks: 16 Integer partition: 16⋅16
| # | serration 1 (quarter sharpness) | block |
| 16 | (0, 0, 0, 0) | [0, 3, 12, 15, 48, 51, 60, 63, 192, 195, 204, 207, 240, 243, 252, 255] |
| 16 | (1, 0, 0, 0) | [1, 2, 13, 14, 49, 50, 61, 62, 193, 194, 205, 206, 241, 242, 253, 254] |
| 16 | (0, 1, 0, 0) | [4, 7, 8, 11, 52, 55, 56, 59, 196, 199, 200, 203, 244, 247, 248, 251] |
| 16 | (1, 1, 0, 0) | [5, 6, 9, 10, 53, 54, 57, 58, 197, 198, 201, 202, 245, 246, 249, 250] |
| 16 | (0, 0, 1, 0) | [16, 19, 28, 31, 32, 35, 44, 47, 208, 211, 220, 223, 224, 227, 236, 239] |
| 16 | (1, 0, 1, 0) | [17, 18, 29, 30, 33, 34, 45, 46, 209, 210, 221, 222, 225, 226, 237, 238] |
| 16 | (0, 1, 1, 0) | [20, 23, 24, 27, 36, 39, 40, 43, 212, 215, 216, 219, 228, 231, 232, 235] |
| 16 | (1, 1, 1, 0) | [21, 22, 25, 26, 37, 38, 41, 42, 213, 214, 217, 218, 229, 230, 233, 234] |
| 16 | (0, 0, 0, 1) | [64, 67, 76, 79, 112, 115, 124, 127, 128, 131, 140, 143, 176, 179, 188, 191] |
| 16 | (1, 0, 0, 1) | [65, 66, 77, 78, 113, 114, 125, 126, 129, 130, 141, 142, 177, 178, 189, 190] |
| 16 | (0, 1, 0, 1) | [68, 71, 72, 75, 116, 119, 120, 123, 132, 135, 136, 139, 180, 183, 184, 187] |
| 16 | (1, 1, 0, 1) | [69, 70, 73, 74, 117, 118, 121, 122, 133, 134, 137, 138, 181, 182, 185, 186] |
| 16 | (0, 0, 1, 1) | [80, 83, 92, 95, 96, 99, 108, 111, 144, 147, 156, 159, 160, 163, 172, 175] |
| 16 | (1, 0, 1, 1) | [81, 82, 93, 94, 97, 98, 109, 110, 145, 146, 157, 158, 161, 162, 173, 174] |
| 16 | (0, 1, 1, 1) | [84, 87, 88, 91, 100, 103, 104, 107, 148, 151, 152, 155, 164, 167, 168, 171] |
| 16 | (1, 1, 1, 1) | [85, 86, 89, 90, 101, 102, 105, 106, 149, 150, 153, 154, 165, 166, 169, 170] |
# Boolf prop/3-ary/serration 2 (half sharpness)

Number of blocks: 4 Integer partition: 4⋅64
| # | serration 2 (half sharpness) | block |
| 64 | (0, 0) | [0, 3, 5, 6, 9, 10, 12, 15, 48, 51, 53, 54, 57, 58, 60, 63, 80, 83, 85, 86, 89, 90, 92, 95, 96, 99, 101, 102, 105, 106, 108, 111, 144, 147, 149, 150, 153, 154, 156, 159, 160, 163, 165, 166, 169, 170, 172, 175, 192, 195, 197, 198, 201, 202, 204, 207, 240, 243, 245, 246, 249, 250, 252, 255] |
| 64 | (1, 0) | [1, 2, 4, 7, 8, 11, 13, 14, 49, 50, 52, 55, 56, 59, 61, 62, 81, 82, 84, 87, 88, 91, 93, 94, 97, 98, 100, 103, 104, 107, 109, 110, 145, 146, 148, 151, 152, 155, 157, 158, 161, 162, 164, 167, 168, 171, 173, 174, 193, 194, 196, 199, 200, 203, 205, 206, 241, 242, 244, 247, 248, 251, 253, 254] |
| 64 | (0, 1) | [16, 19, 21, 22, 25, 26, 28, 31, 32, 35, 37, 38, 41, 42, 44, 47, 64, 67, 69, 70, 73, 74, 76, 79, 112, 115, 117, 118, 121, 122, 124, 127, 128, 131, 133, 134, 137, 138, 140, 143, 176, 179, 181, 182, 185, 186, 188, 191, 208, 211, 213, 214, 217, 218, 220, 223, 224, 227, 229, 230, 233, 234, 236, 239] |
| 64 | (1, 1) | [17, 18, 20, 23, 24, 27, 29, 30, 33, 34, 36, 39, 40, 43, 45, 46, 65, 66, 68, 71, 72, 75, 77, 78, 113, 114, 116, 119, 120, 123, 125, 126, 129, 130, 132, 135, 136, 139, 141, 142, 177, 178, 180, 183, 184, 187, 189, 190, 209, 210, 212, 215, 216, 219, 221, 222, 225, 226, 228, 231, 232, 235, 237, 238] |
# Boolf prop/3-ary/serration size

This is the lowest degree, for which the serration is a vector with only zeros. See degree 3 (sharpness), degree 2 (half sharpness) and degree 1 (quarter sharpness).
Number of blocks: 5 Integer partition: 1⋅1 + 1⋅15 + 1⋅48 + 1⋅64 + 1⋅128
| # | serration size | block |
| 1 | 0 | [0] |
| 128 | 4 | |
| 15 | 1 | [3, 12, 15, 48, 51, 60, 63, 192, 195, 204, 207, 240, 243, 252, 255] |
| 48 | 2 | [5, 6, 9, 10, 53, 54, 57, 58, 80, 83, 85, 86, 89, 90, 92, 95, 96, 99, 101, 102, 105, 106, 108, 111, 144, 147, 149, 150, 153, 154, 156, 159, 160, 163, 165, 166, 169, 170, 172, 175, 197, 198, 201, 202, 245, 246, 249, 250] |
| 64 | 3 | [17, 18, 20, 23, 24, 27, 29, 30, 33, 34, 36, 39, 40, 43, 45, 46, 65, 66, 68, 71, 72, 75, 77, 78, 113, 114, 116, 119, 120, 123, 125, 126, 129, 130, 132, 135, 136, 139, 141, 142, 177, 178, 180, 183, 184, 187, 189, 190, 209, 210, 212, 215, 216, 219, 221, 222, 225, 226, 228, 231, 232, 235, 237, 238] |
# French/Numbers
## Les Chiffres (numbers) 21-61

| number | in French | pronunciations | number | in French | pronunciation |
| 21 | vingt-et-un | van-teh-un | 30 | trente | tront |
| 22 | vingt-deux | van-de | 32 | trente-deux | tront-deux |
| 23 | vingt-trois | vant-trwah | 40 | quarante | karont |
| 24 | vingt-quatre | vant-katr | 41 | quarante-et-un | karont-teh-un |
| 25 | vingt-cinq | vant-sank | 42 | quarante-deux | karont-de |
| 26 | vingt-six | vant-sees | 50 | cinquante | sakote |
| 27 | vingt-sept | vant-set | 51 | cinquante-et-un | sankont-teh-un |
| 28 | vingt-huit | van-tweet | 60 | soixante | swasont |
| 29 | vingt-neuf | vant-nuf | 61 | soixante-et-un | swasont-teh-un |

## Les dizaines 10-100

| number | in French | pronounciation | number | in French | pronounciation |
| 10 | dix | dees | 60 | soixante | swasont |
| 20 | vingt | van | 70 | soixante-dix | swasont-dees |
| 30 | trente | tront | 80 | quatre-vingts | katr-van |
| 40 | quarante | karont | 90 | quatre-vingt-dix | katr-van-dees |
| 50 | cinquante | sankont | 100 | cent | son |

## Les Chiffres (numbers) 1-100

| number | in French | pronunciation |
| 1 | un | un |
| 2 | deux | de |
| 3 | trois | trwah |
| 4 | quatre | katr |
| 5 | cinq | sank |
| 6 | six | sees |
| 7 | sept | set |
| 8 | huit | wheet |
| 9 | neuf | nuf |
| 10 | dix | dees |
| 11 | onze | onze |
| 12 | douze | dooze |
| 13 | treize | trez |
| 14 | quatorze | katorz |
| 15 | quinze | kenz |
| 16 | seize | seiz |
| 17 | dix-sept | dees-set |
| 18 | dix-huit | dees-weet |
| 19 | dix-neuf | dees-nuf |
| 20 | vingt | vent |
| 21 | vingt-et-un | van-teh-un |
| 22 | vingt-deux | van-duh |
| 2_ | vingt-_ | van-turwah |
| 30 | trente | tront |
| 31 | trente-et-un | tront-teh-un |
| 32 | trente-deux | tront-duh |
| 3_ | trente-_ | tront-_ |
| 40 | quarante | karont |
| 41 | quarante-et-un | karont-teh-un |
| 42 | quarante-deux | karont-duh |
| 4_ | quarante-_ | karont-_ |
| 50 | cinquante | sankont |
| 51 | cinquante-et-un | sankont-teh-un |
| 52 | cinquante-deux | sankont-duh |
| 5_ | cinquante-_ | sankont-_ |
| 60 | soixante | swasont |
| 61 | soixante-un | swasont-un |
| 62 | soixante-deux | swasont-de |
| 6_ | soixante-_ | swasont-_ |
| 70 | soixante-dix | swasont-dees |
| 71 | soixante-onze | swasont-onze |
| 72 | soixante-douze | swasont-dooze |
| 7_ | soixante-_ | swasont-_ |
| 80 | quatre-vingt | katr-van |
| 81 | quatre-vingt-un | katr-van-un |
| 82 | quatre-vingt-deux | katr-van-de |
| 8_ | quatre-vingt-_ | katr-van-_ |
| 90 | quatre-vingt-dix | katr-van-dees |
| 91 | quatre-vingt-onze | katr-van-onze |
| 92 | quatre-vingt-douze | katr-van-douze |
| 9_ | quatre-vingt-_ | katr-van-_ |
| 100 | cent | son |
# Boolf prop/3-ary/family size

Number of blocks: 4 Integer partition: 1⋅2 + 1⋅14 + 1⋅56 + 1⋅184
| # | felicity | family size | block |
| 2 | 0 | 1 | [0, 255] |
| 184 | 3 | 8 | |
| 56 | 2 | 4 | [3, 5, 6, 9, 10, 12, 17, 18, 20, 24, 33, 34, 36, 40, 48, 63, 65, 66, 68, 72, 80, 95, 96, 111, 119, 123, 125, 126, 129, 130, 132, 136, 144, 159, 160, 175, 183, 187, 189, 190, 192, 207, 215, 219, 221, 222, 231, 235, 237, 238, 243, 245, 246, 249, 250, 252] |
| 14 | 1 | 2 | [15, 51, 60, 85, 90, 102, 105, 150, 153, 165, 170, 195, 204, 240] |
# Boolf prop/3-ary/gravity

Number of blocks: 5 Integer partition: 2⋅1 + 1⋅6 + 1⋅30 + 1⋅218
| # | gravity | legion weight | legion faction | block |
| 1 | ∞ | ∞ | ∞ | [0] |
| 1 | 3 | 8 | 255 1 Ж 255 | [1] |
| 6 | 2 | 4 | 15 17 Ж 15 | [2, 3, 4, 5, 16, 17] |
| 30 | 1 | 2 | 3 85 Ж 3 | [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, 32, 33, 34, 35, 48, 49, 50, 51, 64, 65, 68, 69, 80, 81, 84, 85] |
| 218 | 0 | 1 | 1 255 Ж 1 | |
# Boolf prop/3-ary/depth

Number of blocks: 5 Integer partition: 2⋅1 + 1⋅14 + 1⋅56 + 1⋅184
| # | depth | cohort weight | block |
| 1 | ∞ | ∞ | [0] |
| 1 | 3 | 8 | [1] |
| 14 | 2 | 4 | [2, 3, 4, 5, 8, 9, 16, 17, 32, 33, 64, 65, 128, 129] |
| 56 | 1 | 2 | [6, 7, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, 24, 25, 34, 35, 36, 37, 40, 41, 48, 49, 50, 51, 66, 67, 68, 69, 72, 73, 80, 81, 84, 85, 96, 97, 104, 105, 130, 131, 132, 133, 136, 137, 144, 145, 152, 153, 160, 161, 164, 165, 192, 193, 194, 195] |
| 184 | 0 | 1 | |
# Boolf prop/3-ary/legion

Number of blocks: 9 Integer partition: 2⋅1 + 3⋅2 + 3⋅10 + 1⋅218
| # | legion | legion set | block |
| 1 | ∞ | N | [0] |
| 1 | 255 1 Ж 255 | {0, 1, 2, 3, 4, 5, 6, 7} | [1] |
| 2 | 85 3 Ж 85 | {0, 2, 4, 6} | [2, 3] |
| 2 | 51 5 Ж 51 | {0, 1, 4, 5} | [4, 5] |
| 10 | 17 15 Ж 17 | {0, 4} | [6, 7, 8, 9, 10, 11, 12, 13, 14, 15] |
| 2 | 15 17 Ж 15 | {0, 1, 2, 3} | [16, 17] |
| 10 | 5 51 Ж 5 | {0, 2} | [18, 19, 32, 33, 34, 35, 48, 49, 50, 51] |
| 10 | 3 85 Ж 3 | {0, 1} | [20, 21, 64, 65, 68, 69, 80, 81, 84, 85] |
| 218 | 1 255 Ж 1 | {0} | |
# Boolf prop/3-ary/cohort

Number of blocks: 17 Integer partition: 2⋅1 + 7⋅2 + 7⋅8 + 1⋅184
| # | cohort | cohort set | block |
| 1 | ∞ | N | [0] |
| 1 | 255 1 Ж 255 | {0, 1, 2, 3, 4, 5, 6, 7} | [1] |
| 2 | 85 3 Ж 85 | {0, 2, 4, 6} | [2, 3] |
| 2 | 51 5 Ж 51 | {0, 1, 4, 5} | [4, 5] |
| 8 | 17 15 Ж 17 | {0, 4} | [6, 7, 10, 11, 12, 13, 14, 15] |
| 2 | 153 7 Ж 153 | {0, 3, 4, 7} | [8, 9] |
| 2 | 15 17 Ж 15 | {0, 1, 2, 3} | [16, 17] |
| 8 | 5 51 Ж 5 | {0, 2} | [18, 19, 34, 35, 48, 49, 50, 51] |
| 8 | 3 85 Ж 3 | {0, 1} | [20, 21, 68, 69, 80, 81, 84, 85] |
| 184 | 1 255 Ж 1 | {0} | |
| 8 | 9 119 Ж 9 | {0, 3} | [24, 25, 136, 137, 144, 145, 152, 153] |
| 2 | 165 19 Ж 165 | {0, 2, 5, 7} | [32, 33] |
| 8 | 33 95 Ж 33 | {0, 5} | [36, 37, 132, 133, 160, 161, 164, 165] |
| 8 | 129 127 Ж 129 | {0, 7} | [40, 41, 72, 73, 96, 97, 104, 105] |
| 2 | 195 21 Ж 195 | {0, 1, 6, 7} | [64, 65] |
| 8 | 65 63 Ж 65 | {0, 6} | [66, 67, 130, 131, 192, 193, 194, 195] |
| 2 | 105 23 Ж 105 | {0, 3, 5, 6} | [128, 129] |
# Boolf prop/3-ary/cohort faction

Number of blocks: 9 Integer partition: 2⋅1 + 1⋅2 + 2⋅6 + 1⋅8 + 2⋅24 + 1⋅184
| # | cohort faction | block |
| 1 | ∞ | [0] |
| 1 | 255 1 Ж 255 | [1] |
| 6 | 15 17 Ж 15 | [2, 3, 4, 5, 16, 17] |
| 24 | 3 85 Ж 3 | [6, 7, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, 34, 35, 48, 49, 50, 51, 68, 69, 80, 81, 84, 85] |
| 6 | 153 7 Ж 153 | [8, 9, 32, 33, 64, 65] |
| 184 | 1 255 Ж 1 | |
| 24 | 9 119 Ж 9 | [24, 25, 36, 37, 66, 67, 130, 131, 132, 133, 136, 137, 144, 145, 152, 153, 160, 161, 164, 165, 192, 193, 194, 195] |
| 8 | 129 127 Ж 129 | [40, 41, 72, 73, 96, 97, 104, 105] |
| 2 | 105 23 Ж 105 | [128, 129] |
# Boolf prop/3-ary/soft legion

Number of blocks: 8 Integer partition: 4⋅2 + 3⋅10 + 1⋅218
| # | soft legion | soft legion set | block |
| 2 | 255 1 Ж 255 | {0, 1, 2, 3, 4, 5, 6, 7} | [0, 1] |
| 2 | 85 3 Ж 85 | {0, 2, 4, 6} | [2, 3] |
| 2 | 51 5 Ж 51 | {0, 1, 4, 5} | [4, 5] |
| 10 | 17 15 Ж 17 | {0, 4} | [6, 7, 8, 9, 10, 11, 12, 13, 14, 15] |
| 2 | 15 17 Ж 15 | {0, 1, 2, 3} | [16, 17] |
| 10 | 5 51 Ж 5 | {0, 2} | [18, 19, 32, 33, 34, 35, 48, 49, 50, 51] |
| 10 | 3 85 Ж 3 | {0, 1} | [20, 21, 64, 65, 68, 69, 80, 81, 84, 85] |
| 218 | 1 255 Ж 1 | {0} | |
# Boolf prop/3-ary/soft cohort

Number of blocks: 16 Integer partition: 8⋅2 + 7⋅8 + 1⋅184
| # | soft cohort | soft cohort set | block |
| 2 | 255 1 Ж 255 | {0, 1, 2, 3, 4, 5, 6, 7} | [0, 1] |
| 2 | 85 3 Ж 85 | {0, 2, 4, 6} | [2, 3] |
| 2 | 51 5 Ж 51 | {0, 1, 4, 5} | [4, 5] |
| 8 | 17 15 Ж 17 | {0, 4} | [6, 7, 10, 11, 12, 13, 14, 15] |
| 2 | 153 7 Ж 153 | {0, 3, 4, 7} | [8, 9] |
| 2 | 15 17 Ж 15 | {0, 1, 2, 3} | [16, 17] |
| 8 | 5 51 Ж 5 | {0, 2} | [18, 19, 34, 35, 48, 49, 50, 51] |
| 8 | 3 85 Ж 3 | {0, 1} | [20, 21, 68, 69, 80, 81, 84, 85] |
| 184 | 1 255 Ж 1 | {0} | |
| 8 | 9 119 Ж 9 | {0, 3} | [24, 25, 136, 137, 144, 145, 152, 153] |
| 2 | 165 19 Ж 165 | {0, 2, 5, 7} | [32, 33] |
| 8 | 33 95 Ж 33 | {0, 5} | [36, 37, 132, 133, 160, 161, 164, 165] |
| 8 | 129 127 Ж 129 | {0, 7} | [40, 41, 72, 73, 96, 97, 104, 105] |
| 2 | 195 21 Ж 195 | {0, 1, 6, 7} | [64, 65] |
| 8 | 65 63 Ж 65 | {0, 6} | [66, 67, 130, 131, 192, 193, 194, 195] |
| 2 | 105 23 Ж 105 | {0, 3, 5, 6} | [128, 129] |
# Boolf prop/3-ary/soft cohort faction

Number of blocks: 8 Integer partition: 2⋅2 + 2⋅6 + 1⋅8 + 2⋅24 + 1⋅184
| # | soft cohort faction | block |
| 2 | 255 1 Ж 255 | [0, 1] |
| 6 | 15 17 Ж 15 | [2, 3, 4, 5, 16, 17] |
| 24 | 3 85 Ж 3 | [6, 7, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, 34, 35, 48, 49, 50, 51, 68, 69, 80, 81, 84, 85] |
| 6 | 153 7 Ж 153 | [8, 9, 32, 33, 64, 65] |
| 184 | 1 255 Ж 1 | |
| 24 | 9 119 Ж 9 | [24, 25, 36, 37, 66, 67, 130, 131, 132, 133, 136, 137, 144, 145, 152, 153, 160, 161, 164, 165, 192, 193, 194, 195] |
| 8 | 129 127 Ж 129 | [40, 41, 72, 73, 96, 97, 104, 105] |
| 2 | 105 23 Ж 105 | [128, 129] |
# Boolf prop/3-ary/soft gravity

Number of blocks: 4 Integer partition: 1⋅2 + 1⋅6 + 1⋅30 + 1⋅218
| # | soft gravity | soft legion weight | soft legion faction | block |
| 2 | 3 | 8 | 255 1 Ж 255 | [0, 1] |
| 6 | 2 | 4 | 15 17 Ж 15 | [2, 3, 4, 5, 16, 17] |
| 30 | 1 | 2 | 3 85 Ж 3 | [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, 32, 33, 34, 35, 48, 49, 50, 51, 64, 65, 68, 69, 80, 81, 84, 85] |
| 218 | 0 | 1 | 1 255 Ж 1 | |
# Boolf prop/3-ary/soft depth

Number of blocks: 4 Integer partition: 1⋅2 + 1⋅14 + 1⋅56 + 1⋅184
| # | soft depth | soft cohort weight | block |
| 2 | 3 | 8 | [0, 1] |
| 14 | 2 | 4 | [2, 3, 4, 5, 8, 9, 16, 17, 32, 33, 64, 65, 128, 129] |
| 56 | 1 | 2 | [6, 7, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, 24, 25, 34, 35, 36, 37, 40, 41, 48, 49, 50, 51, 66, 67, 68, 69, 72, 73, 80, 81, 84, 85, 96, 97, 104, 105, 130, 131, 132, 133, 136, 137, 144, 145, 152, 153, 160, 161, 164, 165, 192, 193, 194, 195] |
| 184 | 0 | 1 | |
# Vector space with inner product/Theorem of Pythagoras/Fact

Let $V$ be a ${\mathbb {K} }$-vector space, endowed with an inner product $\left\langle -,-\right\rangle$. Let $u,v\in V$ be vectors that are
orthogonal to each other.
    $\Vert {u+v}\Vert ^{2}=\Vert {u}\Vert ^{2}+\Vert {v}\Vert ^{2}\,$
# Euclidean space/Distance to subspace/Orthogonal projection/Fact/Proof

For $u\in U$, we have, due to the Pythagorean theorem,
    $d(v,u)^{2}=d(v,p_{U}(v))^{2}+d(p_{U}(v),u)^{2}\,,$
as $p_{U}(v)-u\in U$ and $v-p_{U}(v)\in U^{\perp }$ are perpendicular to each other. This expression is minimal if and only if $d(p_{U}(v),u)=0$, and this holds if and only if
    $p_{U}(v)=u\,.$
# Euclidean space/Distance to subspace/Inner product/Fact/Proof

Because of fact, we have
    $d(v,U)=d(v,p_{U}(v))\,,$
and, according to fact, we have
    $p_{U}(v)=\sum _{i=1}^{m}\left\langle v,u_{i}\right\rangle u_{i}\,.$
We extend the orthonormal basis to an orthonormal basis $u_{1},\ldots ,u_{n}$ of $V$. Therefore, using the Pythagorean theorem, we have
    ${\begin{aligned}d(v,U)^{2}&=d{\left(v,\sum _{i=1}^{m}\left\langle v,u_{i}\right\rangle u_{i}\right)}^{2}\\&=d{\left(\sum _{i=1}^{n}\left\langle v,u_{i}\right\rangle u_{i},\sum _{i=1}^{m}\left\langle v,u_{i}\right\rangle u_{i}\right)}^{2}\\&=\Vert {\sum _{i=m+1}^{n}\left\langle v,u_{i}\right\rangle u_{i}}\Vert ^{2}\\&=\Vert {v}\Vert ^{2}-\Vert {\sum _{i=1}^{m}\left\langle v,u_{i}\right\rangle u_{i}}\Vert ^{2}\\&=\Vert {v}\Vert ^{2}-\sum _{i=1}^{m}\left\langle v,u_{i}\right\rangle ^{2}.\end{aligned}}$
# R^n/Subspace of axes/Distance to point/Example

Let $J\subseteq \{1,\ldots ,n\}$, and let $U=\langle e_{i},\,i\in J\rangle \subseteq \mathbb {R} ^{n}$ denote the linear subspace spanned by this choice of standard vectors. Let
    $v={\begin{pmatrix}v_{1}\\\vdots \\v_{n}\end{pmatrix}}\,.$
Then, the distance of $v$ to $U$ equals
    $d(v,U)={\sqrt {\sum _{i\not \in J}v_{i}^{2}}}\,.$
The dropped perpendicular foot of $v$ on $U$ is
    $p_{U}(v)={\begin{pmatrix}w_{1}\\\vdots \\w_{n}\end{pmatrix}}\,,$
where
    $w_{i}={\begin{cases}v_{i},\,{\text{ if }}i\in J,\,\\0,\,{\text{ if }}i\notin J\,.\end{cases}}\,$
# Euclidean space/Distance to hyperplane/Linear form/Fact/Proof

Let $a,u_{2},\ldots ,u_{n}$ be an orthonormal basis of $\mathbb {R} ^{n}$, and write
    $v=\lambda a+\sum _{i=2}^{n}c_{i}u_{i}\,.$
Then
    $p_{U}(v)=\sum _{i=2}^{n}c_{i}u_{i}\,,$
and, due to fact, we have
    $d(U,v)=\Vert {v-p_{U}(v)}\Vert =\Vert {\lambda a}\Vert =\vert {\lambda }\vert \Vert {a}\Vert =\vert {\lambda }\vert \,.$
In conjunction with
    $\left\langle a,v\right\rangle =\left\langle a,\lambda a+\sum _{i=2}^{n}c_{i}u_{i}\right\rangle =\left\langle a,\lambda a\right\rangle =\lambda \,$
this yields the result.
# SCCAP/Resources for Dealing with a School Shooting/Psychological First Aid

This page serves as a psychological first aid resource that describes the immediate consequences and coping mechanisms for dealing with a school shooting.
Experiencing a trauma is likely to affect you in many ways. There are the immediate consequences that might include losing people you were close to, such as experiencing new fears, feeling angry, and not knowing what to do now that your life has changed in an important way. People will respond to trauma in different ways and will be affected to different degrees. This depends on a lot of factors, including age, previous experience with trauma, available support, and even biological factors, like hormones. There isn't a "right" or "wrong" way to react. The important thing is to pay attention to changes in your mental and physical health and ask for the help you need.
The effects of trauma aren't always negative - some people will experience what is called "post-traumatic growth," and some people will be relatively unchanged months after the trauma. There is no reason to assume that you will have long-term negative consequences. Getting help early is a good way to reduce the chances of longer term problems. Here are some ways a traumatic event might affect you:

## What to expect

As you are recovering from a traumatic event, you/your caregiver/friends/family may be experiencing various feelings and changes in response to the event. These are normal, and may change as time passes. Here is a summary of responses in the recovery period after a traumatic event:
| | Negative Responses | Positive Responses |
| Cognitive | Confusion, disorientation, worry, unwanted thoughts and images, self-blame | Focused, alert |
| Emotional | Shock, sadness, fear, anger, guilt | Motivated, determined, courageous, optimistic |
| Social | Withdrawal from others, conflict | Increased connectedness, involvement in helping behaviors |
| Physiological | Headache, stomachache, fast heartbeat, trouble sleeping | Increased energy, ready to respond |
# Medicine/First Aid

First aid is any assistance that can be given to a victim before medical help arrives. It can range from putting a bandaid on a cut to administering CPR. Medical emergencies do not occur every day, but being prepared and knowing how to treat them can help you keep your calm in these situations. It is important for all adults to have a functional knowledge of how to handle common medical emergencies.

## Common injuries/emergencies

The most common injuries/emergencies requiring First Aid are:

## How to Make a First Aid Kit

You should keep at least one first aid kit in your home and one in every car you and your family own. Whether it is a bee sting or a cut, a well stocked first aid kit will have all the supplies you need to treat minor ailments. A first aid kit should include:
Make sure that you read the first aid manual and know how to properly use all the items in the kit. Make sure you also train your family in how to use the kit, because you may be the one who needs the treatment! The best place to keep your first aid kit is in your kitchen, since this is where many activities take place. The bathroom is not a good place to keep your kit because the humidity can shorten the shelf life of many items.
A list of emergency numbers can also be useful in case a minor injury or sickness turns into something more serious. An easy template for all your local emergency numbers can be found at KidsHealth.org.

## How Can I Learn More About First Aid?

Many hospitals, health departments, and other health organizations will offer First Aid and CPR classes to its community members. In these classes you will have the opportunity to learn directly from health professionals. You will be able to observe as well as demonstrate First Aid procedures for most of the common injuries/emergencies mentioned earlier. Upon successful completion of the course you will receive a certification in First Aid and CPR. To find out more about First Aid/CPR classes in your area, you can contact your local health department or hospital. You can also visit the American Red Cross website to search for classes in your area.
# Euclidean plane/(4,5)/Distance to 2x-3y is 7/Example

We want to determine in the Euclidean plane the distance between the point $P={\begin{pmatrix}4\\5\end{pmatrix}}$ and the line $G$ given by $2x-3y=7$. The line has the form
    $G={\left\{{\begin{pmatrix}{\frac {7}{2}}\\0\end{pmatrix}}+t{\begin{pmatrix}3\\2\end{pmatrix}}\mid t\in \mathbb {R} \right\}}\,,$
and ${\begin{pmatrix}2\\-3\end{pmatrix}}$ is a vector perpendicular to $G$. We have
    ${\begin{pmatrix}{\frac {7}{2}}\\0\end{pmatrix}}-{\begin{pmatrix}4\\5\end{pmatrix}}={\begin{pmatrix}-{\frac {1}{2}}\\-5\end{pmatrix}}=-{\frac {23}{26}}{\begin{pmatrix}3\\2\end{pmatrix}}+{\frac {14}{13}}{\begin{pmatrix}2\\-3\end{pmatrix}}\,.$
Therefore, the dropped perpendicular foot is
    ${\begin{pmatrix}4\\5\end{pmatrix}}+{\frac {14}{13}}{\begin{pmatrix}2\\-3\end{pmatrix}}={\begin{pmatrix}{\frac {80}{13}}\\{\frac {23}{13}}\end{pmatrix}}\,,$
and the distance is
    ${\frac {14}{13}}{\sqrt {13}}.$
# History of China/Modern China and Beyond: People's Republic of China and the Modern World

This is the third unit of Chinese History
* The People's Republic of China
** Mao Zedong
* One Country, Two Systems
** Taiwan
** Hong Kong
** Macau
* China Today (1989-present)
* The Future of China, the awakening dragon.