recursal/Fanatic-Fandom huggingface dataset:
Collect fossil

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Tips for beating mission:

1. It's preffered to use magnets (from 2-4) to grab the fossil.
2. It's easier to do if you clear area from enemies first.

Things about the mission:

* The fossil is nearly under first ground layer.
* This mission is common on Bug Planet (add link to this).
Bone Eater Plant Nature Demon Knight

**Agaram**

A member of The Dark Army Imagine Corps. He is classified at the infernal level, demon knight class.

## Biography

A demon knight who specializes in manipulating the bone eater plant. He serves in The Order of the Demon Woods, whose patron is the mad kakuja fist koukaku ghoul nature illusion demon god knight.

Agaram is a humanoid demon who is wizened and pale skinned. He has long ears and greasy, scraggly, dirty white hair that reaches his chin. He has a long, big nose and is covered in wrinkles. He has crimson eyes with black vertical slit pupils. He stands hunched over at 5'4. He has gnarled hands with long fingernails. He wears a crimson red with a white leather belt with two bags on either hip.

He is a strange and eccentric demon who is sadistic and cruel. He treats the bone eater plant as his pet and cares for it lovingly. He enjoys seeing the red bloom when it feeds and is gleeful in death. He croons to his plants and sees himself as their parent. He considers himself superior and an arbitrator of life and death. He has self importance on a grand scale and is proud of his powers. When any hurt his "precious babies", he flies into a rage and is murderous and violent.

\*Note: the Bone Eater Plant is a white plant that feeds on bones. It looks like a bone white plant with a segmented carapace that is made of ossified nutrients. It is as hard as bone and has a head shaped like a tulip that hides rows of teeth when it blooms. It can crunch through bone easily. This plant also has tendrils sheathed in a bone-like armor.

## Powers/Abilities/Techniques

Hand to Hand Combat

Can call down the power of higher beings.

Signature Technique: Bone Eater Plant Manipulation - He can throw seeds of the plant from his right belt bag and imbue them with his demonic energy to make them sprout and move the way he wishes.

Wild Bone Eater's Dance - Tendrils of the Bone Eater whip out in all directions.

He can make the bone eater plant bind enemies.

His bone eaters are immune to fire.

Secret Technique: Gardener of Death - His bones have been replaced with the Bone Eater Plant itself. This plant is held in place with his will and does not grow, but becomes harder and more durable the more it is fed. It is currently as hard as steel.

He can combine the bones of the bone eater plant to create stronger structures and attacks.

Fa Jin Master

Forbidden Technique: Father of Doom - Upon death, he can feed all of his demonic energy into the Bone Eater Plant that makes his skeleton. It then grows to its' full size and power and becomes a danger to all around it.

Ultimate Technique: Chosen Bones - At the pouch on his left hip, he keeps seeds covered in the bones of special beings. When the seeds sprout and feed on these bones, they gain special attributes, powers, and abilities based on the bones they eat.

-Alien Bones - Various alien abilities.

**Equipment:**

Bag of the Seeds of the Bone Eater Plant covered in a bone shell.

Demon Stone
Bount Body

**Soul Absorption**: Bounts normally try to absorb a Human's soul after a person dies and is trying to pass on. But they are also capable of taking the souls of those who are living and doing so kills the Human.[31]

* **Immortality**: By absorbing Human souls, a Bount will retain their youth and can potentially live forever.[31]

* **Empowerment**: By taking the souls of living Humans, Bounts gain considerable increased might and unique additional abilities

Master of Hand-to-hand combat

Great Spiritual Power

Enhanced Strength

Enhanced Speed

Enhanced Durability

Enhanced Endurance

Reiatsu Stealth

Spirit Particle Absorption

Doll Attunement

Doll

-Blutkönig (Blood King): nicknamed Vlad. This doll is made out of the blood of the user and usually takes the shape of a vampire like Dracula. When unreleased, the case is a crimson armlet with a sapphire gem in the middle of the inside forearm.

Vlad has enhanced speed, enhanced strength, and can reform no matter what physical attacks attack the body. The more blood that is around, the more it can add to itself. It can fully manipulate blood and attack with whatever fuels the body's imagination. The only weakpoint is a core that will be somewhere in the blood around the size of an egg. This core is bloodred, making it harder to see.

Vampirkonig (Vampire King)-When this ability is activated, the body and Vlad fuse together and make each other stronger than the sum of the parts. The fusion retains the ability of both with greater efficiency and power than before.
XNor Gate

**Deactivates** output only when **exactly one of the two inputs** are active.
(Or, it activates only when both inputs are either active OR inactive).
It's exactly the same as an Xor Gate followed by a Not Gate.

It's known as 'not-exclusive-or', because while an Xor Gate only activate when one OR the other (and NOT both) inputs are active, the XNor will only activate when they are both inactive or both active.

*The name can be very confusing if we think of it as being an exclusive-nor. As a NOR gate is only active when there are no signals, and an XNOR is also active when there are no signals, it is not the best way of looking at it. So, it's much better to remember that it should be called a NXor, but because that's hard to pronounce, some mathematician switched it to be called an XNor.*

## Examples

### When Just One Sensor is Active, start a Jump Thruster

For instance, imagine a SENSE\_LEFT and a SENSE\_RIGHT, and we want to tie a MOVE action, which itself is tied to a Jump Thruster when we see one OR the other sensors (but not both) active. This could be when we want to turn but need to ensure that there is enough space to turn..

If we were using a Thruster we would use an Xor Gate but because a Jump Thruster activates when it stops charging, we want to do the opposite, and this gate does that perfectly for us.

## Tag Conventions

Typical **Output Tag** values are context driven, but are often prefixed with "SAME". Eg, for the inputs "FRONT X" and "BACK X", an XNOR output Tag could be "SAME X", etc.
Boar Demon Knight

**Gorshok**

A member of The Dark Army Imagine Corps. He is classified at the infernal level, demon knight class.

## Biography

A demon knight of the boar demon race. He serves in the legions of the wild beast devouring magic demon supreme god knight as member of The Order of the Demon Beast.

Gorshok is a boar demon with immense power. He is a mighty force and a juggernaut shock trooper of great success. He is a greedy, lustful being who is driven by fleshly desires. He enjoys picking on the weak and smashing apart those he is stronger than.

## Powers/Abilities/Techniques

Immense Strength - He is strong enough to wield his hammer effortlessly and devastate even enemies wearing full armor.

Beast Senses

Hammer Fighting Skills

Hand to Hand Combat Skills

Can call down the power of higher beings.

Secret Technique: Fa Jin

Forbidden Technique: Earth Shaking Step

**Equipment:**

Plate Armor

Immense Hammer - 8 feet long with a head the size of a car. It weighs several tons.

Demon Stone
Celestial Moon Sect

A sect of cultivators created by a female immortal. They strive to perfect their martial arts while bringing life, beauty, and good to the world.

Their symbol is a crescent moon atop a pink background.

**Members:**

Xin Zhao

Ashe

Zhou Zu -Cursed Cultivator (Self Sealed)

Wei Hai -Acid Cultivator

Lin Fei Cheng -Wind Cultivator

Hai Su Liu -Water Cultivator

Wu Shou Dong -Beast Spirit Cultivator

Zheng Dou Ling -Warrior Cultivator

Wang Di -Earth Cultivator

Zhao Ze Bai -Black Arts Cultivator

Mu Biao -Nature Cultivator

Hui Zha Yen -Fire Cultivator (Ascended immortal and leader of a new sect)

Annie (In Training)

Ru Guo -Lightning Cultivator

Du Qi -Poison Cultivator

Huai Bo -Destruction Cultivator (Self Sealed)

Wai Tai Kong -Astral Cultivator

Yi Wei -Doctor Cultivator

Jing Shen -Psychic Cultivator

Xia Yu Ning -Rain Cultivator

Sheng Hua -Sound Cultivator

Zhong Li -Gravity Cultivator

Ah Re -Heat Cultivator
Bloody Arms

A derivation of the Crimson Arms technique. It is one step up from Scarlet Breaker in that it uses a vast amount of demonic energy, but is all compressed into the user's arm. The arms lose their look of armored plating from before and become so dark as to become almost black. They regain the original speed of Crimson Arms and retain half the power of Scarlet Breaker. However, this technique forces a lot of strain onto the user's arms. If it is overused, the user's arms will become crippled for all time.
Belphegor (Celestial)

**Belphegor**

A member of The Dark Army. He is classified at the transcendent celestial level.

He is a general that is a member of The Seven Deadly Sins. He leads The Legion of Sloth.

## Biography

In Christian demonology, **Belphegor** (Hebrew: בעל־פעור) is the Arch-devil. In qabalah, Belphegor is a demon who helps people make discoveries. He seduces people by suggesting to them ingenious inventions that will make them rich, stagnating that which could not be accredited to itself.

Auxiliary Bishop and witch-hunter Peter Binsfeld believed that Belphegor tempts by means of laziness. Also, according to Peter Binsfeld's Classification of Demons, Belphegor is the chief demon of the deadly sin known as Sloth in Christian tradition.

## Powers/Abilities/Techniques

Powers of Sloth

Can bring ideas to others.

Shapeshifting.

Power of Ignorance

Vast Energy Reserves

High Self Healing Abilities

Signature Technique: Celestial Dreamwalking

Secret Technique: Sloth Debt

Forbidden Technique: True Form = He can fight in his true body and maximize the efficiency of his fighting power.

Ultimate Technique: Restful Sleep

**Equipment:**

Stone of the Deep
Call of the Deep

The signature technique of Keana. Her mantle fluctuates and fills with her demonic energy and transforms to have images of the sea within the folds of the cloak. From the folds of the cantle come the spirits and creatures of the ocean to fight and swarm over the enemy. The more and stronger the creatures summoned, the more demonic energy Keana must release in order to stay in control.
Bakura Ryou

**Bakura Ryou**

A member of The Dark Army Imagine Corps. He is classified at the summoner level, legendary duelist class.

## Biography

**Bakura Ryou** ( *Bakura Ryō*), known as **Ryo Bakura** in the Japanese version, is a recurring character in the *Yu-Gi-Oh!* anime. He is a classmate and friend of Yugi Muto.

He is the modern-day counterpart to Bandit King Bakura.

Yami Bakura is his other self, released from the power of the Millennium Ring. The two rarely directly converse. Unlike the close, cooperative relationship between Yugi Muto and Yami Yugi, Yami Bakura is a malignant entity that completely suppresses Bakura's spirit when he takes over him.

## Infinite Reality System

**Assets:**

Ghoul with an Appetite

Sleeping LionGandora, the Dragon of Destruction

White Knight Castle

The Dark -Hex Sealed Fusion

Hibikime

Mega Thunderball

Mirror Mail

Yami

Sealing Waterfall

Sonic Maid

Thing in the Mud

Ansatsu

Musician King

Mirror Wall

Beast King of the Swamp

Mirror Force

Dian Keto the Cure Master

Shifting Shadows

Gravi Rush Dragon

Water Omotics

Greed Cards

Number 73: Abyss Splash

Elemental Hero Clayman

Tarkaminos

Sword Hunter

Giant Orc

Granmarg the Rock Monarch

Zombie Master

Shadow Delver

Allure Queen LV 3

Berserk Gorilla

Cloudian Nimbusman

Laser Spyder

Legendary Ocean

Dust Barrier

White Knight Swordsman

Beast Jaws

Cloudian Poison Cloud

Hibikime

Kaminari Attack

Crystal Beast Cards

Iron Chain Dragon

Sacred Forest

Anti Magic Arrows

**Equipment:**

Millennium Ring
Aviso de Tenis 2: De vuelta a la escuela

## Reparto

### Voces adicionales

#### TV

* Zoe Mendoza
* Pablo Gutierrez
* Lorenzo Alessandro Castañeda
* Claudia Bustamante
* Jared García
* Ivanna Palacios
* José Aarón Vázquez

* Verónica Gregorio
* Abraham Viloria
* Jesús Abraham López y López
* Claudia Cárdenas
* Claudia Mayta
* Óscar Eduardo Hernández Rabañales
* Juan Carlos López
* Joshuan Humberto Burelo Tut
* Ximena Durán
* Óscar Rolando Sosa Marroquín
* Juan Carlos Pérez
* Zaori Gutiérrez
* Karla Fernán

* Juan Carlos Zúñiga Caba
* Ángel David Pacheco Muñoz
* Kaori Cruz
* Jonathan Hiroshy Choc Fuentes
* Valeria Mejía
* Xóchitl Gayosso
* Miguel Adolfo Sacayom Obispo
* Carina Álvarez

* Marco Tulio Alejandro Tax Colop
* Marely Hernández
* Alexis Iván Sis Bín
* Ximena Montoya
* Triana Carolina
* Carlos Armando Camey González
* Diana Lisseth
* Gabriela Gedeón
* Carlos Antonio Castillo Miranda

* Celso Gutiérrez
* Jesmi Castillo
* Gaby Waywa
* Nelson Argumedo
* Carla Copa
Bat Demon

**Bat Demon**

A member of The Dark Army Imagine Corps. He is classified at the infernal level, lower class.

## Biography

A demon of the bat race. He serves the little demon queen's little queendom.

A bat demon tends to have low intelligence. They are savage things that are barely intelligent enough to clothe themselves. They can only communicate amongst each other and live in great colonies. However, they are fierce, loyal, and very dangerous creatures.

## Behind the scenes

Claws

Flight

Echolocation

Fangs

Nocturnal

**Weaknesses:**

Light, Sound Attacks
Blue Eye of Negation

An eye inherited by Caeruleum from his father. It is in the middle of his forehead, and is vertical. When it opens, blue light streams from it and negates all supernatural abilities under the level of an imperial celestial. However, it costs an immense amount of mana to use and cannot be used for a long time after it has been opened. Caeruleum can only use it to negate for 3 seconds.
Blood Armor Vampire Demon Knight

**Mura**

A member of The Dark Army Imagine Corps. He is classified at the infernal level, demon prince class.

At one point, he became the commander of The Order of Nosferatu after he became a demon lord. However, then a count and led the Vampire Demon Clan. He is currently a demon prince who leads all the demonic knight orders.

## Biography

A demon knight who serves the Blazing Blood Demon Martial Great Celestial in The Order of Nosferatu. Mura is a 6'5 humanoid demon with darkened, pale skin, red eyes, and a fanged mouth. He has white hair that is very modern and faded with stringy, wavy hair that reaches his nose. He has a cruel face with a pointed chin and nose, and a perpetual sneer that is only broken by scowls. He wears a loose fitting black tunic and black breeches with black shoes.

Mura is a cocky, arrogant demon who takes to the front lines to show off his power. He believes himself unbreakable and unbeatable and will prove it by dominating those weaker than himself. He easily loses his cool and is easily provoked. He is prideful in his powerful abilities and will use every opportunity to show them off.

## Powers/Abilities/Techniques

Vampire Physiology

Enhanced Strength

Enhanced Agility

Enhanced Speed

Enhanced Senses

Enhance Durability

Night Vision

Fa Jin

Hemokinesis

Blood Godslayer Magic

Blood Devilslayer Magic

Can call down power from his ultimate master.

Can bestow his power unto others.

Self Cleanse Ability

Signature Technique: Blood Armor

Blood Gathering Technique

Secret Technique: Blood Armor Mode Change - Mura has several mode specializations he can use to modify his Blood Armor to maximum effect.

* Titan - Increases the arm size to give it maximum strength. Defense is increased in this form, but speed drops down drastically.

* Flash - Trims down the bulk and strength of the armor and allows the user to move with blinding speed.

* Swordmaster - Trims some bulk down from the armor and creates blades jutting from the armor. This allows him to use devastating sword techniques.

* Arsenal - Creates a massive turret jutting from the armor's back as well as as many turrets and launchers as the user wants. This allows the user to launch devastating ranged blood attacks.
* Juggernaut - Maximizes the defense and thickness of the armor's plates making it night impregnable.
* Airmaster - Grants wings on the back of the armor and trims down some excess weight, allowing the user to fly.
* Stealth - Makes the armor skintight and grants them slight physical enhancements, but almost no defense.

Crimson Avatar

Red Shade

Forbidden Technique: Dark Red - Mura's use of the technique takes the form of a massive blood dragon.

Ultimate Technique: Crimson Colossus

**Weaknesses:**

Sunlight

Holy

**Equipment:**

Blood Stone
Blazing Palm Banasura

The ultimate technique of the Blazing Sun Palace. It summons the image of a banasura with blazing sun energy formed from the user's qi. This banasura can attack with 1,000 arms at high speed and devastating power while providing extra range from its' large size. Each strike is further augmented by blazing sun qi.
Aviso de Tenis

**Aviso de Tenis** es una película de animación por producción en la futuro

## Reparto (México)

### **Diamond Films]]**

* Lisa Arias
* Jared Mendoza
* Zoe Mora
* Carla Copa
* Jahel Mendoza
* José Gabino Ángeles
* Annie Reyes
* José Aarón Vázquez
* Juan Alberto Domínguez
* Ana Karen Martínez
* Randy Díaz Hernández
* Gabriela Garay
* Alicia Mendoza
* Esmeralda Mendoza
* Karina
Anexo:Especiales de MacKenzie Blie

**Especiales de Mackenzie Blie (serie).**

## **La chica en la escuela**

### **Voces adicionales**

* Valentina Rossi
* Mara Bermúdez
* Agustina Celeste
* María Elena Merida

## **La maldición de Cleopatra**

### **Voces adicionales**

**redoblaje original**

* Daiana Hernández
* Andrea Hernández
* Sofīa Navarro
* Triana Carolina
* Luciana Díaz
* Marcela Nerlys
* Stefani Rosalba
* Valentina Chacón

**redoblaje 2**

* Gabriela Belén
* Marisol Dionicio
* Arelys González
* María Sarmiento
* Kelly Viera
* Stefany Vargas
* Valentina Tenorio
Armour of Reason

Known in the legends of his Legion as the "ever-reforged" armour, it was said that Roboute Guilliman himself had this set of Artificer Armour remade and adapted countless times if ever a flaw or weakness was discovered in battle, and at various times the artisanship both of Mars and his fellow Primarchs Vulkan and Perturabo influenced its design in the days before the sundering of the Imperium in the fires of the Heresy.

It is an ascended celestial level armor that has the ability to repair itself and improve itself each time it is breached.
Diamonds in the Rough

**Diamonds in the Rough** is the first episode in the series.

## Plot synopsis

*Here's where you put the description of the plot. Doesn't have to be very detailed. A couple sentences will be enough for starters. You can expand it later.*

## Notes & trivia

* *This is where you can put notes and trivia, anything that is interesting or strange can be included here*
ANDA (estudio)

**Animación y Narración de Doblaje y Artísticas** es una empresa de doblaje estadounidense

## Actores de doblaje

**Actores de doblaje**

* Juan Andrés Velasco
* Javier Ruiz
* Roberto Álvarez
* Fransisco Mendoza
* Willy Herrera
* Anthony Jesús
* Jorge Luis Urango
* Roberto Lema
* Humberto Alexander

**Actrices de doblaje**

* Maricar Noboa
* María José Esquitin
* Erika Gómez
* Anabel Maldonaldo
* Estefanía Ordaz
* María Salazar
* Ruth Tarrillo
* Keiko Aquino
* Annie Escobar
* Estrellita González
* Diana Lisseth
* Arianna Lisseth
* Carmen Citllali
* Esmeralda Mendoza
Andrea Herrera

## Información artística

### Ha realizado doblaje en

 Argentina
 (2036-presente)
 México
 (2042)

**Andrea Herrera** es una actriz de doblaje argentina.

## Fimografia

### **Películas animadas**

* Halloween High: Electrizadas - Cleo de Zile
* Bella y los mágicos sirenas - Chelsea
* Halloween High: Rápidos sin race - Cleo de Zile
* Halloween High: Felices vacaciones a los chicas - Cleo de Zile
* Halloween High: Terror de viernes - Cleo de Zile
* Monster High: Retorno a la gran arrefice monstruosa - Clio de Nile

### **Cortos animadas**

* Aventuras del escuadrón de monstruos: Edición de fiesta - Cleo de Zile
Propositions, théorèmes, lemmes, corollaires et axiomes

Dans un cours de mathématiques, on distingue généralement différents types d’assertions, afin de
leur donner plus ou moins d’importance. Le choix de désigner une assertion comme un théorème par
exemple, dépendra de l’auteur et du cours en question.

## Proposition

Quand une assertion a le mérite de devoir être retenue, on l’appelle **proposition**.

## Théorème

Quand une proposition est très importante, on l’appelle **théorème**.

## Lemme

Quand une assertion est utile pour démontrer une proposition, on l’appelle **lemme**.

## Corollaire

Quand une proposition découle directement d’une autre proposition, on préfère l’appeler
**corollaire**.

## Axiome

On appelle **axiome** toute assertion que l’on considère vraie sans la démontrer.
Implication à droite

## Proposition

* Soient

P

{\displaystyle P}

,

Q

{\displaystyle Q}

 et

R

{\displaystyle R}

 trois assertions.

L'implication suivante est vraie:

(

(
P
∧
Q
)
⇒
R

⇓

P
⇒
(
Q
⇒
R
)

)

{\displaystyle \left(\begin{array}{c}(P\wedge Q)\Rightarrow R\\\Downarrow\\P\Rightarrow(Q\Rightarrow R)\end{array}\right) }

## Notation utilisées sur cette page

* ∧

{\displaystyle \wedge}

: la conjonction
* ⇒

{\displaystyle \Rightarrow}

: l'implication (entre deux assertions)
Win Token

**Win Token** is one of the two hosts of Battle Of Objects Meta. He is childish, stubborn, and quick-witted.

## Biography

In Diamonds in the Rough, he auditions to be the host of a new object show. Even though he failed his audition, his extreme persistence forced Briefcase to hire him alongside Diamond.

## Behind the scenes

Before the reboot of the show, Win Token appeared as the only host in Battle Of Objects, Battle Of Objects Two, and Actual Battle Of Objects Two. In his first appearance in Battle Of Objects, he is armless. As the series progresses, he develops more of an edge, becoming similar to his appearance in BOOM.
Battle Of Objects Two (Old Season 2)

**Season 2** of Battle Of Objects consists of 6 episodes.

The season premiered on October 15, 2015.

## Plot overview

Most of the contestants from season 1 return to compete in another season. A few new faces are also introduced. Blueberry struggles to prove his innocence, until episode 4, when a new investigation finds Blueberry innocent, and Win Token guilty instead. Boxy finds a time crystal, allowing him to know when the timeline is altered. He finds that Win Token's boss Casey is using time control powers to manipulate everyone. Casey then wakes up in an alternate reality where nothing makes sense. At the end of the episode, it is revealed that the whole season had been a dream, which started right before the court case in episode 13. Blueberry is now found innocent rather then guilty. Since the season had not been real, a new season would then start: Actual Battle Of Objects Two.

## Trailer

*You can replace this text with a video trailer using our "Add a video" tool. Or you can remove this section entirely.*

## Episodes

* 1: Unexpected Surprise
* 2: Fundraisin
* 3: Just Plane Awful
* 4: Land Of The Free (previously named Not Long Ago, In A Galaxy Not Far Away)
* 5: Parabox
* 6: Parallel Universe
Diamond

**Diamond** is the sassy, apathetic host of BOOM.

## Biography

In episode 1, she passed the hosting audition, becoming the host of the show. When she was given a tour of the valley where the competition takes place, Win Token shows up uninvited and becomes the other host of the show. Diamond and Win Token cannot cooperate with each other,and she is forced to work with him because she does not have a valid complaint against him.

## Behind the scenes

Diamond was originally going to be an amethyst stone, but was then changed to a diamond stone.
Beach Ball

{{infobox character

```
Beach Ball       =

```

}}
**Character name** is...

This first section is called a lead. You'll want to fill it with the most important aspects of the character's life. Think about a newspaper article. Put the best stuff up front!

Use this section to give greater details about the character's life.

## Behind the scenes

If there's anything you'd like to say that's not derived from episodes, but maybe comes from interviews that actors or crew gave, use this section.
Battle Of Objects (Old Season 1)

**Season 1** of Battle Of Objects consists of 13 episodes.

The season premiered on July 1, 2013.

## Plot overview

20 inanimate objects join an object show hosted by Win Token to escape boredom and to win a million dollars. They compete in deadly challenges, one contestant being voted out of the losing team by the viewers each episode, with a few exceptions. As the season progresses, we learn more about Win Token and Blueberry. Blueberry has been caught cheating, and Win Token sues him, the season ending with Blueberry and Boomerang being arrested, the latter escaping.

## Trailer

*You can replace this text with a video trailer using our "Add a video" tool. Or you can remove this section entirely.*

## Episodes

* 1a: Popping By (previously named The Meeting)
* 1b: Staying Up (previously named Firework Extravagant)
* 2: Token Risks (previously named What's A Win Token?)
* 3: Kicking It Off (previously named Football Frenzy)
* 4: Armed and Ready (previously named The Unrevealed Truth)
* 5: Swallowed in Thought
* 6: Frozens Aisle (previously named Ice Age Again)
* 7: Bring Me Home (previously named Migration Route)
* 8: A Rival's Arrival (previously named Just Like in the Game)
* 9: Technical Difficulties
* 10: A Rival's Return (previously named De Ja Vu)
* 11: I Got Your Backstory (previously named The Life Of A Win Token)
* 12a: Hidden Feelings (previously named Not Quite Normal
* 12b: Oh Grape, Even More Trouble
* 13: Court An Ending
Basketball

**Basketball** is big and apathetic. He is teased for his own weight, and he is not offended, he simply admits that they are right.

## Biography

In Diamonds in the Rough, he forms an alliance with Fedora.

## Behind the scenes

If there's anything you'd like to say that's not derived from episodes, but maybe comes from interviews that actors or crew gave, use this section.
Disjonction de deux assertions

## Définition

Soient

P

{\displaystyle P}

 et

Q

{\displaystyle Q}

 deux assertions.

On définit alors une nouvelle assertion, appelée **disjonction** de P et de Q, notée

P
∨
Q

{\displaystyle P\vee Q}

, et définie par la table de vérité suivante:

Il suffit donc qu'une seule des deux assertions soit vraie, pour que

P
∨
Q

{\displaystyle P\vee Q}

 soit vraie.

## Autre notation

* P
 ou
Q

{\displaystyle P\text{ ou }Q}

## Exemple

L'assertion

(
x
<
0
)
∨
(
# x
0
)

{\displaystyle (x<0)\vee(x=0)}

, qui s'écrit généralement

x
≤
0

{\displaystyle x \le 0}

.

## Remarque

Dans le langage courant, le "ou" est **exclusif**: au restaurant, si on vous demande "entrée ou désert", vous ne pouvez prendre qu’un seul des deux. L’assertion P∨Q du langage courant n’est pas vraie quand P et Q sont vraies simultanément. Ainsi, il y aurait un F à la place du V dans la dernière ligne du tableau si on prenait le "ou" exclusif du langage courant. Le "ou" des mathématiques est dit **inclusif**.
Battle Of Objects Meta (Season 1)

**Season 1** of Battle Of Objects Meta consists of 0 episodes that have been released.

The season is expected to premiere on January 1, 2018.

## Plot overview

*This is where you write about the general story in the season so far. A few sentences will be enough for starters. You can expand this section later.*

## Trailer

*You can replace this text with a video trailer using our "Add a video" tool. Or you can remove this section entirely.*

## Episodes

* 1: Diamonds in the Rough
* 2: Relationship Goals
Union et inclusion

## Proposition

* Soit

C

{\displaystyle C}

 un pseudo-ensemble. Soit

A

{\displaystyle A}

 une partie de

C

{\displaystyle C}

.
* Soit

D

{\displaystyle D}

 un pseudo-ensemble. Soit

B

{\displaystyle B}

 une partie de

D

{\displaystyle D}

.

On a alors

A
∪
B
⊆
C
∪
D

{\displaystyle A\cup B\subseteq C\cup D}

.

## Démonstration

Soit

x

{\displaystyle x}

 un pseudo-ensemble.

Si

x
∈
A
∪
B

{\displaystyle x\in A\cup B}

, alors d'après cette proposition,

x
∈
A

{\displaystyle x \in A}

 ou

x
∈
B

{\displaystyle x \in B}

.

Or,

A
⊆
C

{\displaystyle A\subseteq C}

 donc

x
∈
C

{\displaystyle x \in C}

 ou

x
∈
B

{\displaystyle x \in B}

.

Mais

B
⊆
D

{\displaystyle B\subseteq D}

 donc

x
∈
C

{\displaystyle x \in C}

 ou

x
∈
D

{\displaystyle x \in D}

.

Donc toujours d'après cette proposition,

x
∈
C
∪
D

{\displaystyle x\in C\cup D}

.

Ainsi,

∀
x
,

(

x
∈
A
∪
B
⇒
x
∈
C
∪
D

)

{\displaystyle \forall x,{\Big (}x\in A\cup B\Rightarrow x\in C\cup D{\Big )}}

.

Donc

A
∪
B
⊆
C
∪
D

{\displaystyle A\cup B\subseteq C\cup D}

.

**CQFD**.

## Notations utilisées sur cette page

* ⇒

{\displaystyle \Rightarrow ~~}

l'implication
* ∈

{\displaystyle \in~~}

l'appartenance
* ∀

{\displaystyle \forall~~}

pour tout
* ⊆

{\displaystyle \subseteq~~}

l'inclusion
* A
∪
B

{\displaystyle A\cup B~~}

l'union
Première loi de De Morgan (version assertions)

## Proposition

* Soient

P

{\displaystyle P}

 et

Q

{\displaystyle Q}

 deux assertions.

L'équivalence suivante est vraie:

¬
(
P
∧
Q
)

⟺

(
¬
P
∨
¬
Q
)

{\displaystyle \neg(P\wedge Q)~~\iff~~(\neg P\vee\neg Q)}

## Vocabulaire

Cette équivalence est appelée "**Première loi de De Morgan**"

## Notations utilisées sur cette page

* ¬

{\displaystyle \neg}

: la négation
* ∧

{\displaystyle \wedge}

: la conjonction
* ∨

{\displaystyle \vee}

: la disjonction
* ⟺

{\displaystyle \iff}

: l'équivalence
Quantificateur universel

## Définition

* Soit

P

{\displaystyle P}

 une assertion dépendant d'un paramètre.
* Soit

E

{\displaystyle E}

 un pseudo-ensemble.

(
1
)

{\displaystyle (1)}

 L'assertion "Pour tout pseudo-ensemble x, on a P(x) vraie" se note

∀
x
,
P
(
x
)

{\displaystyle \forall x, P(x)}

.

(
2
)

{\displaystyle (2)}

 L'assertion "Pour tout élément x de E, on a P(x) vraie" se note

∀
x
∈
E
,
P
(
x
)

{\displaystyle \forall x\in E,P(x)}

.

## Propriétés

* Négation des quantificateurs
* L'assertion

∀
x
∈
E
,
P
(
x
)

{\displaystyle \forall x\in E,P(x)}

 peut se réécrire

∀
x
,

(

x
∈
E
⇒
P
(
x
)

)

{\displaystyle \forall x,\Big(x\in E\Rightarrow P(x)\Big)}

## Remarque

* Il y a aussi le quantificateur existentiel.
Axiome de remplacement des assertions équivalentes

## Axiome

* Soient

P

{\displaystyle P}

 et

Q

{\displaystyle Q}

 deux assertions.

On admet le fait que si

P

⟺

Q

{\displaystyle P\iff Q}

 est démontrée, alors on pourra remplacer

P

{\displaystyle P}

 par

Q

{\displaystyle Q}

 partout où

P

{\displaystyle P}

 apparait. De même, on pourra remplacer

Q

{\displaystyle Q}

 par

P

{\displaystyle P}

 partout où

Q

{\displaystyle Q}

 apparaît.

## Notations utilisées sur cette page

* ⟺

{\displaystyle \iff}

: l'équivalence
Chapitre 1: Éléments de logique

Retour au cours

## Assertions

* Assertion
* Assertion dépendant d'un paramètre
* Propositions, théorèmes, lemmes, corollaires et axiomes
* Table de vérité d'une assertion

## Assertions usuelles

* Négation d'une assertion
* Conjonction de deux assertions
* Disjonction de deux assertions
* Implication
* Démonstration
* Équivalence entre deux assertions
* Axiome de remplacement des assertions équivalentes
* La table de vérité de l'équivalence

## Propriétés de la négation

* Involution de la négation
* Négation à gauche
* Négation à droite

## Propriétés de la conjonction

* Idempotence de la conjonction
* Principe de non-contradiction
* Symétrie de la conjonction
* Associativité de la conjonction
* Distributivité de la conjonction sur la disjonction
* Distributivité de la disjonction sur la conjonction
* Première loi de De Morgan (version assertions)
* Deuxième loi de De Morgan (version assertions)
* Les neutres de la conjonction
* Les neutres de la conjonction 2
* Les absorbants de la conjonction
* Perte d'information en conjonction

## Propriétés de la disjonction

* Idempotence de la disjonction
* Principe du tiers exclu
* Symétrie de la disjonction
* Associativité de la disjonction
* Les neutres de la disjonction
* Les absorbants de la disjonction
* Perte d'information en disjonction

## Propriétés basiques de l'implication

* Réflexivité de l'implication
* Transitivité de l'implication
* Affirmation de l'implication
* Négation de l'implication
* Implication à droite
* Implication et conjonction
* Implication et disjonction

## Propriétés de l'équivalence

* Réflexivité de l'équivalence entre deux assertions
* Symétrie de l'équivalence entre deux assertions
* Transitivité de l'équivalence entre assertions
* Équivalence et perte d'information

## Contraposée et réciproque d'une implication

* Contraposée d'une implication
* Contraposée et équivalence
* Réciproque d'une implication

## Raisonnements

* Modus Ponens
* Raisonnement par série d'équivalences
* Raisonnement par série d'implications
* Raisonnement par double implications
* Raisonnement par contraposition
* Raisonnement par l'absurde

## Propriétés élaborées de l'implication

* Distributivité de l'implication
* Fusion d'implications
* Fusion d'implications et réflexivité
Négation à droite

## Axiome

* Soient

P

{\displaystyle P}

,

Q

{\displaystyle Q}

 et

R

{\displaystyle R}

 trois assertions.

L'implication suivante est vraie:

(
[
P
∧
Q
]
⇒
R
)

⇒

(
P
⇒
[
¬
Q
∨
R
]
)

{\displaystyle ([P\wedge Q]\Rightarrow R)~~\Rightarrow~~(P\Rightarrow[\neg Q\vee R])}

## Notations utilisées sur cette page

* ∧

{\displaystyle \wedge}

: la conjonction
* ∨

{\displaystyle \vee}

: la disjonction
* ⇒

{\displaystyle \Rightarrow}

: l'implication
* ¬

{\displaystyle \neg}

: la négation
Ensemble vide et quantificateurs

## Proposition

* Soit

P

{\displaystyle P}

 une assertion dépendant d'un paramètre.

(
1
)

{\displaystyle (1)}

 Il est toujours vrai de dire que

∀
x
∈
∅
,
P
(
x
)

{\displaystyle \forall x\in\emptyset,P(x)}

.

(
2
)

{\displaystyle (2)}

 Il est toujours faux de dire que

∃
x
∈
∅
,
P
(
x
)

{\displaystyle \exists x\in\emptyset,P(x)}

.

## Idée de preuve

La proposition

(
1
)

{\displaystyle (1)}

 se montre en se rappelant que pour un pseudo-ensemble

E

{\displaystyle E}

, l'assertion

∀
x
∈
E
,
P
(
x
)

{\displaystyle \forall x\in E,P(x)}

 peut se réécrire

∀
x
,

(

x
∈
E
⇒
P
(
x
)

)

.

{\displaystyle \forall x,\Big(x\in E\Rightarrow P(x)\Big).}

Ainsi, l'assertion

∀
x
∈
∅
,
P
(
x
)

{\displaystyle \forall x\in\emptyset,P(x)}

 peut se réécrire

∀
x
,

(

x
∈
∅
⇒
P
(
x
)

)

{\displaystyle \forall x,\Big(x\in\emptyset\Rightarrow P(x)\Big)}

.

Il suffit alors de se souvenir que comme

x
∈
∅

{\displaystyle x\in\emptyset}

 est toujours faux (puisque l'ensemble vide est vide), l'implication

x
∈
∅
⇒
P
(
x
)

{\displaystyle x\in\emptyset\Rightarrow P(x)}

 est vraie.

La proposition

(
2
)

{\displaystyle (2)}

 se montre en remarquant que sa négation est

∀
x
,
¬
P
(
x
)

{\displaystyle \forall x,\neg P(x)}

 qui est toujours vraie, d'après

(
1
)

{\displaystyle (1)}

.

## Notations utilisées sur cette page

* ⇒

{\displaystyle \Rightarrow}

 l'implication
* ∈

{\displaystyle \in}

 l'appartenance
* ∃

{\displaystyle \exists}

 il existe
* ∀

{\displaystyle \forall}

 pour tout
Union de deux pseudo-ensembles

## Définition

* Soient

A

{\displaystyle A}

 et

B

{\displaystyle B}

 des pseudo-ensembles.

On appelle **union** de

A

{\displaystyle A}

 et de

B

{\displaystyle B}

 l'ensemble

⋃
{
A
;
B
}

{\displaystyle \bigcup \{A;B\}}

.

On le nota

A
∪
B

{\displaystyle A \cup B}

# A
{
1
;
2
}

{\displaystyle A=\{1;2\}}

 et

# B
{
2
;
3
;
4
}

{\displaystyle B=\{2;3;4\}}

, alors

A
∪
# B
⋃
{
A
;
B
# }
⋃
{
{
1
;
2
}
;
{
2
;
3
;
4
}
# }
{
1
;
2
;
3
;
4
}

{\displaystyle A\cup B=\bigcup \{A;B\}=\bigcup \{\{1;2\};\{2;3;4\}\}=\{1;2;3;4\}}

.

## Propriétés de l'union

### Seule

* Autre écriture de l'union

x
∈
A
∪
B

{\displaystyle x\in A\cup B}

 si et seulement si

x
∈
A

{\displaystyle x \in A}

 ou

x
∈
B

{\displaystyle x \in B}

* Inclusions immédiates dans l'union

A
⊆
A
∪
B

{\displaystyle A\subseteq A\cup B}

 et

B
⊆
A
∪
B

{\displaystyle B\subseteq A\cup B}

* Commutativité de l'union

A
∪
# B
B
∪
A

{\displaystyle A \cup B = B \cup A}

* Associativité de l'union

(
A
∪
B
)
∪
# C
A
∪
(
B
∪
C
)

{\displaystyle (A \cup B) \cup C = A \cup (B \cup C)}

* Idempotence de l'union

A
∪
# A
A

{\displaystyle A\cup A = A}

* Absorbants de l'union

Si

A
⊆
Ω

{\displaystyle A\subseteq\Omega}

, alors

A
∪
# Ω
Ω

{\displaystyle A\cup \Omega =\Omega }

* Neutre de l'union

A
∪
# ∅
A

{\displaystyle A \cup \emptyset = A}

* Union et inclusion

Si

A
⊆
C

{\displaystyle A\subseteq C}

 et

B
⊆
D

{\displaystyle B\subseteq D}

, alors

A
∪
B
⊆
C
∪
D

{\displaystyle A\cup B\subseteq C\cup D}

* Minimalité de l'union

Si

A
⊆
Ω

{\displaystyle A\subseteq\Omega}

 et

B
⊆
Ω

{\displaystyle B\subseteq \Omega }

, alors

A
∪
B
⊆
Ω

{\displaystyle A\cup B\subseteq \Omega }

## Notations utilisées sur cette page

* =

{\displaystyle =}

 l'égalité
* ⋃

{\displaystyle \bigcup}

 la réunion
Contraposée d'une implication

## Définition

* Soient

P

{\displaystyle P}

 et

Q

{\displaystyle Q}

 deux assertions.

L'implication

¬
Q
⇒
¬
P

{\displaystyle \neg Q\Rightarrow\neg P}

 est appelée **contraposée** de l'implication

P
⇒
Q

{\displaystyle P\Rightarrow Q}

.

## Exemples

* La contraposée de

(
x
≥
1
)
⇒
(

x

2

≥
1
)

{\displaystyle (x\geq 1)\Rightarrow(x^{2}\geq 1)}

 est

(

x

2

<
1
)
⇒
(
x
<
1
)

{\displaystyle (x^{2}<1)\Rightarrow(x<1)}

.

* La contraposée de "*Si c'est une pomme, alors c'est un fruit*" est "*Si ce n'est pas un fruit, alors ce n'est pas une pomme*".

## Attention

Il ne faut pas confondre la contraposée avec la réciproque (sera abordé plus loin dans le cours).

## Notations utilisées sur cette page

* ¬

{\displaystyle \neg}

: la négation
* ⇒

{\displaystyle \Rightarrow}

: l'implication
Fusion d'implications

## Proposition

* Soient

A

{\displaystyle A}

,

B

{\displaystyle B}

,

P

{\displaystyle P}

 et

Q

{\displaystyle Q}

 quatre assertions.

L'implication suivante est vraie

(

[

A
⇒
B

]

∧

[

P
⇒
Q

]

)

⇒

(

[

A
∧
P

]

⇒

[

B
∧
Q

]

)

{\displaystyle {\Big (}{\Big [}A\Rightarrow B{\Big ]}\wedge {\Big [}P\Rightarrow Q{\Big ]}{\Big )}\Rightarrow {\Big (}{\Big [}A\wedge P{\Big ]}\Rightarrow {\Big [}B\wedge Q{\Big ]}{\Big )}}

## Démonstration

Raisonnons par série d'implications. Cliquez sur les ☑ pour obtenir les justifications.

(
1
)

(

A
⇒
B

)

∧

(

P
⇒
Q

)

{\displaystyle (1)~~~~{\Big (}A\Rightarrow B{\Big )}\wedge {\Big (}P\Rightarrow Q{\Big )}}

☑☑

⇒
(
2
)

(

¬
A
∨
B

)

∧

(

¬
P
∨
Q

)

{\displaystyle \Rightarrow (2)~~~~{\Big (}\neg A\vee B{\Big )}\wedge {\Big (}\neg P\vee Q{\Big )}}

☑

⇒
(
3
)

(
¬
A
∨
B
)
∧
¬
P

∨

(
¬
A
∨
B
)
∧
Q

{\displaystyle \Rightarrow (3)~~~~{\begin{array}{c}(\neg A\vee B)\wedge \neg P\\\vee \\(\neg A\vee B)\wedge Q\end{array}}}

☑☑

⇒
(
4
)

(
¬
A
∧
¬
P
)
∨
(
B
∧
¬
P
)

∨

(
¬
A
∧
Q
)
∨
(
B
∧
Q
)

{\displaystyle \Rightarrow (4)~~~~{\begin{array}{c}(\neg A\wedge \neg P)\vee (B\wedge \neg P)\\\vee \\(\neg A\wedge Q)\vee (B\wedge Q)\end{array}}}

☑☑

⇒
(
5
)

¬
A
∨
(
B
∧
¬
P
)

∨

(
¬
A
∧
Q
)
∨
(
B
∧
Q
)

{\displaystyle \Rightarrow (5)~~~~{\begin{array}{c}\neg A\vee (B\wedge \neg P)\\\vee \\(\neg A\wedge Q)\vee (B\wedge Q)\end{array}}}

☑☑

⇒
(
6
)

¬
A
∨
¬
P

∨

(
¬
A
∧
Q
)
∨
(
B
∧
Q
)

{\displaystyle \Rightarrow (6)~~~~{\begin{array}{c}\neg A\vee \neg P\\\vee \\(\neg A\wedge Q)\vee (B\wedge Q)\end{array}}}

☑☑

⇒
(
7
)

¬
A
∨
¬
P

∨

¬
A
∨
(
B
∧
Q
)

{\displaystyle \Rightarrow (7)~~~~{\begin{array}{c}\neg A\vee \neg P\\\vee \\\neg A\vee (B\wedge Q)\end{array}}}

☑☑

⇒
(
8
)
¬
A
∨
¬
P
∨
(
B
∧
Q
)

{\displaystyle \Rightarrow (8)\neg A\vee \neg P\vee (B\wedge Q)}

☑☑

⇒
(
9
)

¬
(
A
∧
P
)
∨
(
B
∧
Q
)

{\displaystyle \Rightarrow (9)~~~~\neg (A\wedge P)\vee (B\wedge Q)}

☑

⇒
(
10
)

(
A
∧
P
)
⇒
(
B
∧
Q
)

{\displaystyle \Rightarrow (10)~~~~(A\wedge P)\Rightarrow (B\wedge Q)}

Par transitivité de l'implication, on a donc montré que

(
1
)
⇒
(
10
)

{\displaystyle (1)\Rightarrow (10)}

.

**CQFD**.

## Notations utilisées sur cette page

* ¬

{\displaystyle \neg}

: la négation
* ∧

{\displaystyle \wedge}

: la conjonction
* ∨

{\displaystyle \vee}

: la disjonction
* ⇒

{\displaystyle \Rightarrow}

: l'implication
Raisonnement par l'absurde

## Définition

* Soit

Q

{\displaystyle Q}

 une assertion.

**Raisonner par l'absurde** pour prouver

Q

{\displaystyle Q}

, c'est démontrer une implication

¬
Q
⇒
¬
P

{\displaystyle \neg Q\Rightarrow\neg P}

 où

P

{\displaystyle P}

 est une assertion que l'on sait vraie.

Par contraposition, on a donc montré l'implication

P
⇒
Q

{\displaystyle P\Rightarrow Q}

. Or,

P

{\displaystyle P}

 est vraie. Donc par Modus Ponens,

Q

{\displaystyle Q}

 est vraie.

## Notations utilisées sur cette page

* ¬

{\displaystyle \neg}

 la négation
* ⇒

{\displaystyle \Rightarrow}

 l'implication
Transitivité de l'équivalence entre assertions

## Proposition

* Soient

P

{\displaystyle P}

,

Q

{\displaystyle Q}

 et

R

{\displaystyle R}

 trois assertions.

L'implication suivante est vraie:

(

[
P
⇔
Q
]
∧
[
Q
⇔
R
]

)

⇒

(
P
⇔
R
)

{\displaystyle \Big([P\Leftrightarrow Q]\wedge[Q\Leftrightarrow R]\Big)~~\Rightarrow~~(P\Leftrightarrow R)}

## Vocabulaire

* Cette implication est appelée "**transitivité** de l'équivalence entre deux assertions".
* On dit que l'équivalence entre deux assertions est **transitive**.

## Notations utilisées sur cette page

* ∧

{\displaystyle \wedge}

: la conjonction
* ⇒

{\displaystyle \Rightarrow}

: l'implication
* ⟺

{\displaystyle \iff}

: l'équivalence
Contraposée et équivalence

## Proposition

* Soient

P

{\displaystyle P}

 et

Q

{\displaystyle Q}

 deux assertions.

L'équivalence suivante est vraie:

(
P
⇒
Q
)

⟺

(
¬
Q
⇒
¬
P
)

{\displaystyle (P\Rightarrow Q)\iff (\neg Q\Rightarrow \neg P)}

Autrement dit: une implication et sa contraposée sont toujours équivalentes.

## Notations utilisées sur cette page

* ¬

{\displaystyle \neg}

 la négation
* ⇒

{\displaystyle \Rightarrow}

 l'implication
* ⟺

{\displaystyle \iff}

 l'équivalence
* ∨

{\displaystyle \vee}

 la disjonction
Négation de l'implication

## Proposition

* Soient

P

{\displaystyle P}

 et

Q

{\displaystyle Q}

 deux assertions.

L'équivalence

¬
(
P
⇒
Q
)

⟺

(
P
∧
¬
Q
)

{\displaystyle \neg (P\Rightarrow Q)\iff (P\wedge \neg Q)}

 est vraie.

## Démonstration

D'après la table de vérité de l'équivalence, il suffit de montrer que

¬
(
P
⇒
Q
)

{\displaystyle \neg (P\Rightarrow Q)}

 et

P
∧
¬
Q

{\displaystyle P\wedge \neg Q}

 ont la même table de vérité.

Tout d'abord, la table de vérité de

¬
(
P
⇒
Q
)

{\displaystyle \neg (P\Rightarrow Q)}

Ensuite, la table de vérité de

P
∧
¬
Q

{\displaystyle P\wedge \neg Q}

.

Les deux tables sont bien identiques: il y a donc équivalence.

**CQFD**.

## Notations utilisées sur cette page

* ¬

{\displaystyle \neg}

: la négation
* ⇒

{\displaystyle \Rightarrow}

: l'implication
* ⟺

{\displaystyle \iff}

: l'équivalence
* ∧

{\displaystyle \wedge}

: Conjonction de deux assertions
Assertion

## Définition

Une phrase dont le contenu porte sur des mathématiques, et qui peut être ou bien **vraie**, ou bien **fausse**, est appelée une **assertion**.

## Exemples

* L'assertion

2
≤
3

{\displaystyle 2\leq3}

 est une assertion vraie.

* L'assertion

2
# 2
5

{\displaystyle 2+2=5}

 est une assertion fausse.

## Vocabulaire

* Dans certains cours, le mot **prédicat** est utilisé à la place du mot assertion.
* Dans certains cours, le mot **proposition** est utilisé à la place du mot assertion. Ça ne sera pas le cas ici, car ce mot sera utilisé pour désigner des assertions un peu particulières.

## Remarques

* Dans ce cours, quand on fera des raisonnements, on n’utilisera que des assertions, de manière à soit avoir quelque chose de vrai, soit quelque chose de faux.
Deuxième loi de De Morgan (version assertions)

## Proposition

* Soient

P

{\displaystyle P}

 et

Q

{\displaystyle Q}

 deux assertions.

L'équivalence suivante est vraie:

¬
(
P
∨
Q
)

⟺

(
¬
P
∧
¬
Q
)

{\displaystyle \neg(P\vee Q)~~\iff~~(\neg P\wedge\neg Q)}

## Vocabulaire

Cette équivalence est appelée "**Deuxième loi de De Morgan**".

## Notations utilisées sur cette page

* ¬

{\displaystyle \neg}

: la négation
* ∧

{\displaystyle \wedge}

: la conjonction
* ∨

{\displaystyle \vee}

: la disjonction
* ⟺

{\displaystyle \iff}

: l'équivalence
Négation de l'existence dans un ensemble

## Proposition

Soient

P

{\displaystyle P}

 une assertion dépendant d'un paramètre, et

E

{\displaystyle E}

 un ensemble.

L'équivalence suivante est vraie:

¬
[
∃
x
∈
E
,
P
(
x
)
]

⟺

∀
x
∈
E
,
¬
P
(
x
)

{\displaystyle \neg[\exists x\in E,P(x)]\iff\forall x\in E,\neg P(x)}

## Démonstration

Raisonnons par série d'équivalences.

(
1
)

¬
[
∃
x
∈
E
,
P
(
x
)
]

{\displaystyle (1)~~~~\neg[\exists x\in E,P(x)]}

On applique à

(
1
)

{\displaystyle (1)}

 la définition du quantificateur existentiel pour obtenir

(
2
)

¬
[
∃
x
,
(
x
∈
E
∧
P
(
x
)
)
]

{\displaystyle (2)~~~~\neg[\exists x, (x\in E\wedge P(x))]}

On construit les équivalences suivantes (cliquez sur les ☑ pour les explications):

* P
(
x
)

⟺

¬
[
¬
P
(
x
)
]

.

{\displaystyle P(x)\iff\neg[\neg P(x)]~~~~.}

 (☑)
* (
x
∈
E
∧
P
(
x
)
)

⟺

(
x
∈
E
∧
¬
[
¬
P
(
x
)
]
)

.

{\displaystyle (x\in E\wedge P(x))\iff(x\in E\wedge\neg[\neg P(x)])~~~~.}

 (☑)
* ∃
x
,
(
x
∈
E
∧
P
(
x
)
)

⟺

∃
x
,
(
x
∈
E
∧
¬
[
¬
P
(
x
)
]
)

.

{\displaystyle \exists x, (x\in E\wedge P(x))\iff \exists x,(x\in E\wedge\neg[\neg P(x)])~~~~.}

 (☑)

On applique à

(
2
)

{\displaystyle (2)}

 la proposition Négation et équivalence pour obtenir

(
3
)

¬
[
∃
x
,
(
x
∈
E
∧
¬
[
¬
P
(
x
)
]
)
]

{\displaystyle (3)~~~~\neg[\exists x,(x\in E\wedge\neg[\neg P(x)])]}

On applique à

(
3
)

{\displaystyle (3)}

 la définition du quantificateur existentiel pour obtenir

(
4
)

¬
[
∃
x
∈
E
,
¬
[
¬
P
(
x
)
]
]

{\displaystyle (4)~~~~\neg[\exists x\in E,\neg[\neg P(x)]]}

On applique à

(
4
)

{\displaystyle (4)}

 la définition du quantificateur universel pour obtenir

(
5
)

∀
x
∈
E
,
¬
P
(
x
)

{\displaystyle (5)~~~~\forall x\in E, \neg P(x)}

**CQFD**.

## Notations utilisées sur cette page

* ¬

{\displaystyle \neg}

: la négation
* ∃

{\displaystyle \exists}

: il existe
* ∈

{\displaystyle \in}

: l'appartenance
* ⟺

{\displaystyle \iff}

: l'équivalence
* ∀

{\displaystyle \forall}

: pour tout
* ∧

{\displaystyle \wedge}

: la conjonction
Intersection d'un pseudo-ensemble

## Définition

* Soit

E

{\displaystyle E}

 un pseudo-ensemble **non vide**.
* Soit

A
∈
E

{\displaystyle A\in E}

.

D'après cette proposition, le pseudo-ensemble suivant ne dépend pas du choix de

A

{\displaystyle A}

.

On appelle **intersection de**

E

{\displaystyle E}

 le pseudo-ensemble

{

x
∈
A

∀
B
∈
E
,
x
∈
B

}

{\displaystyle \big\{x\in A~|~\forall B\in E,x\in B\big\}}

, que l'on va noter

⋂
E

{\displaystyle \bigcap E}

.

C'est l'intersection de tous les éléments de

E

{\displaystyle E}

.

## Notations utilisées sur cette page

* ∈

{\displaystyle \in~~}

l'appartenance
* ∀

{\displaystyle \forall~~}

pour tout
Symétrie de la disjonction

## Proposition

* Soient

P

{\displaystyle P}

 et

Q

{\displaystyle Q}

 deux assertions.

L'équivalence suivante est vraie:

(
P
∨
Q
)

⟺

(
Q
∨
P
)

{\displaystyle (P\vee Q)\iff (Q\vee P)}

## Vocabulaire

* Cette équivalence s'appelle **la symétrie de la disjonction**.

* On dit que **la disjonction est symétrique**.

## Notations utilisées sur cette page

* ∨

{\displaystyle \vee}

: la disjonction
* ⟺

{\displaystyle \iff}

: l'équivalence