Crystal’s Website 💜

Welcome to Crystal’s Cozy Nook!

@@ -35,19 +35,19 @@ I’m over the moon that you’re here, adding even more sweetness to th

Articles ( NEW !!!! )

Discord : an internet cancer Sun Sep 10 15:25:22 2023

root@localhost $ whoami

-My name is Crystal, I’m a <insert time since 09/12/2005> years old transfem from Algeria. Well I’m also pansexual, anarcho-communist and absurdist! I speak 4 languages (Kabyle/Tamazight Arabic French & English) if this info is of any use for you !! +I’m Crystal, your 18 y/o Algerian anarchist transfem, who’s also a geek for some reason…and surprisingly still alive!!!

@@ -80,23 +80,23 @@ My GNUPG (GPG) public key ./src/txt/pubkey.asc

Sign my Guestbook (External website warning)

Want to leave a message, opinion, review or a salty insult ? Be sure to Sign my Guestbook then, it takes two seconds but it will mean the world to me !!!

Blinkies

@@ -132,27 +132,27 @@ Want to leave a message, opinion, review or a salty insult ? Be sure to Sign my

My banner

If you enjoyed my website, you could link me on your personal website using this banner. If you don’t want to, then no pressure 💜 I still love you and I hope that this small shrine of mine will impress you in the future!!!

Webrings & Links (JAVASCRIPT WARNING)!!

Misc :

My University notes

@@ -161,7 +161,7 @@ If you enjoyed my website, you could link me on your personal website using this

Author: Crystal

Created: 2023-10-01 Sun 22:08

Created: 2023-10-16 Mon 21:51

\ No newline at end of file diff --git a/src/org/index.org b/src/org/index.org index 305f519..7d1b35f 100755 --- a/src/org/index.org +++ b/src/org/index.org @@ -20,7 +20,7 @@ I'm over the moon that you're here, adding even more sweetness to this corner! * Articles ( NEW !!!! ) - *[[./articles/discord.html][Discord : an internet cancer]]* /Sun Sep 10 15:25:22 2023/ * root@localhost $ whoami -My name is *Crystal*, I'm a years old transfem from Algeria. Well I'm also pansexual, anarcho-communist and absurdist! I speak 4 languages (Kabyle/Tamazight Arabic French & English) if this info is of any use for you !! +I'm *Crystal*, your 18 y/o Algerian anarchist transfem, who's also a geek for some reason...and surprisingly still alive!!! My current setup is : - Primary OS: *OpenBSD -current* diff --git a/src/org/uni_notes/algebra1.org b/src/org/uni_notes/algebra1.org index 4afc389..1126423 100755 --- a/src/org/uni_notes/algebra1.org +++ b/src/org/uni_notes/algebra1.org @@ -150,7 +150,7 @@ not(*P ∨ Q*) ⇔ P̅ ∧ Q̅ *A rule I really like to use here is: Break and Invert. Basically you break the bar into the three characters of the propositions, so you get not(P) not(∧ or ∨) /NOT AN ACTUAL MATH WRITING. DONT USE IT ANYWHERE ELSE OTHER THAN YOUR BRAIN/ and not(Q)* *** *Transitivity*: -[(P ⇒ Q) (Q ⇒ R)] ⇔ P ⇒ R +[(P ⇒ Q) AND (Q ⇒ R)] ⇔ P ⇒ R *** *Contraposition*: (P ⇒ Q) ⇔ (Q̅ ⇒ P̅) @@ -162,7 +162,7 @@ not(*P ∨ Q*) ⇔ P̅ ∧ Q̅ and -(Q̅ ⇒ Q) is true +(P̅ ⇒ Q) is true then diff --git a/uni_notes/algebra.html b/uni_notes/algebra.html index 27c8d76..9323119 100755 --- a/uni_notes/algebra.html +++ b/uni_notes/algebra.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Algebra 1 @@ -47,13 +47,13 @@

Algebra 1

Contenu de la Matiére

Rappels et compléments (11H)

Logique mathématique et méthodes du raisonnement mathématique
Ensembles et Relations

Structures Algébriques (11H)

Groupes et morphisme de groupes
Anneaux et morphisme d’anneaux

Polynômes et fractions rationnelles

Notion du polynôme à une indéterminée á coefficients dans un anneau
Opérations Algébriques sur les polynômes

Premier cours : Logique mathématique et méthodes du raisonnement mathématique Sep 25 :

Let P Q and R be propositions which can either be True or False. And let’s also give the value 1 to each True proposition and 0 to each false one.

@@ -438,13 +438,13 @@ A proposition is equivalent to another only when both of them have the same v Note: P implying Q is equivalent to P̅ implying Q̅, or: (P ⇒ Q) ⇔ (P̅ ⇒ Q̅)

-
-
Properties:
-
+
+
Properties:
+

-
-
Absorption:
-
+
+
Absorption:
+

(P ∨ P) ⇔ P
@@ -454,9 +454,9 @@ A proposition is equivalent to another only when both of them have the same v

-
-
Commutativity:
-
+
+
Commutativity:
+

(P ∧ Q) ⇔ (Q ∧ P)
@@ -466,9 +466,9 @@ A proposition is equivalent to another only when both of them have the same v

-
-
Associativity:
-
+
+
Associativity:
+

P ∧ (Q ∧ R) ⇔ (P ∧ Q) ∧ R
@@ -478,9 +478,9 @@ P ∨ (Q ∨ R) ⇔ (P ∨ Q) ∨ R

-
-
Distributivity:
-
+
+
Distributivity:
+

P ∧ (Q ∨ R) ⇔ (P ∧ Q) ∨ (P ∧ R)
@@ -490,9 +490,9 @@ P ∨ (Q ∧ R) ⇔ (P ∨ Q) ∧ (P ∨ R)

-
-
Neutral element:
-
+
+
Neutral element:
+

We define proposition T to be always true and F to be always false
@@ -506,9 +506,9 @@ P ∨ F ⇔ P

-
-
Negation of a conjunction & a disjunction:
-
+
+
Negation of a conjunction & a disjunction:
+

Now we won’t use bars here because my lazy ass doesn’t know how, so instead I will use not()!!!
@@ -526,25 +526,25 @@ not(P ∨ Q) ⇔ P̅ ∧ Q̅

-
-
Transitivity:
-
+
+
Transitivity:
+

-[(P ⇒ Q) (Q ⇒ R)] ⇔ P ⇒ R +[(P ⇒ Q) AND (Q ⇒ R)] ⇔ P ⇒ R

-
-
Contraposition:
-
+
+
Contraposition:
+

(P ⇒ Q) ⇔ (Q̅ ⇒ P̅)

-
-
God only knows what this property is called:
-
+
+
God only knows what this property is called:
+

If
@@ -558,7 +558,7 @@ and

-(Q̅ ⇒ Q) is true +(P̅ ⇒ Q) is true

@@ -571,17 +571,17 @@ Q is always true

-
-
Some exercices I found online :
-
+
+
Some exercices I found online :
+

-
-
USTHB 2022/2023 Section B :
-
+
+
USTHB 2022/2023 Section B :
+

-
Exercice 1: Démontrer les équivalences suivantes:
-
+
Exercice 1: Démontrer les équivalences suivantes:
+

(P ⇒ Q) ⇔ (Q̅ ⇒ P̅) @@ -635,8 +635,8 @@ Literally the same as above 🩷

-
Exercice 2: Dire si les propositions suivantes sont vraies ou fausses, et les nier:
-
+
Exercice 2: Dire si les propositions suivantes sont vraies ou fausses, et les nier:
+

∀x ∈ ℝ ,∃y ∈ ℝ*+, tels que e^x = y @@ -769,13 +769,13 @@ y + x < 8

-
-
2éme cours Oct 2
-
+
+
2éme cours Oct 2
+

-
-
Quantifiers
-
+
+
Quantifiers
+

A propriety P can depend on a parameter x
@@ -791,8 +791,8 @@ A propriety P can depend on a parameter x

-
Example
-
+
Example
+

P(x) : x+1≥0
@@ -803,13 +803,13 @@ P(X) is True or False depending on the values of x

-
-
Proprieties
-
+
+
Proprieties
+

-
Propriety Number 1:
-
+
Propriety Number 1:
+

The negation of the universal quantifier is the existential quantifier, and vice-versa :
@@ -820,8 +820,8 @@ The negation of the universal quantifier is the existential quantifier, and vice

-
Example:
-
+
Example:
+

∀ x ≥ 1 x² > 5 ⇔ ∃ x ≥ 1 x² < 5
@@ -829,8 +829,8 @@ The negation of the universal quantifier is the existential quantifier, and vice

-
Propriety Number 2:
-
+
Propriety Number 2:
+

∀x ∈ E, [P(x) ∧ Q(x)] ⇔ [∀ x ∈ E, P(x)] ∧ [∀ x ∈ E, Q(x)]
@@ -841,8 +841,8 @@ The propriety “For any value of x from a set E , P(x) and Q(x)” is e

-
Example :
-
+
Example :
+

P(x) : sqrt(x) > 0 ; Q(x) : x ≥ 1
@@ -860,8 +860,8 @@ P(x) : sqrt(x) > 0 ; Q(x) : x ≥ 1

-
Propriety Number 3:
-
+
Propriety Number 3:
+

∃ x ∈ E, [P(x) ∧ Q(x)] ⇒ [∃ x ∈ E, P(x)] ∧ [∃ x ∈ E, Q(x)]
@@ -872,8 +872,8 @@ P(x) : sqrt(x) > 0 ; Q(x) : x ≥ 1

-
Example of why it’s NOT an equivalence :
-
+
Example of why it’s NOT an equivalence :
+

P(x) : x > 5 ; Q(x) : x < 5
@@ -886,8 +886,8 @@ Of course there is no value of x such as its inferior and superior to 5 at the s

-
Propriety Number 4:
-
+
Propriety Number 4:
+

[∀ x ∈ E, P(x)] ∨ [∀ x ∈ E, Q(x)] ⇒ ∀x ∈ E, [P(x) ∨ Q(x)]
@@ -901,16 +901,16 @@ Of course there is no value of x such as its inferior and superior to 5 at the s

-
-
Multi-parameter proprieties :
-
+
+
Multi-parameter proprieties :
+

A propriety P can depend on two or more parameters, for convenience we call them x,y,z…etc

-
Example :
-
+
Example :
+

P(x,y): x+y > 0
@@ -926,8 +926,8 @@ P(-2,-1) is a False one

-
WARNING :
-
+
WARNING :
+

∀x ∈ E, ∃y ∈ F , P(x,y)
@@ -943,8 +943,8 @@ Are different because in the first one y depends on x, while in the second one,

-
Example :
-
+
Example :
+

∀ x ∈ ℕ , ∃ y ∈ ℕ y > x -–— True
@@ -958,8 +958,8 @@ Are different because in the first one y depends on x, while in the second one,

-
Proprieties :
-
+
Proprieties :
+

not(∀x ∈ E ,∃y ∈ F P(x,y)) ⇔ ∃x ∈ E, ∀y ∈ F not(P(x,y))

not(∃x ∈ E ,∀y ∈ F P(x,y)) ⇔ ∀x ∈ E, ∃y ∈ F not(P(x,y))
@@ -968,20 +968,20 @@ Are different because in the first one y depends on x, while in the second one,
-
-
Methods of mathematical reasoning :
-
+
+
Methods of mathematical reasoning :
+

-
-
Direct reasoning :
-
+
+
Direct reasoning :
+

To show that an implication P ⇒ Q is true, we suppose that P is true and we show that Q is true

-
Example:
-
+
Example:
+

Let a,b be two Real numbers, we have to prove that a² + b² = 1 ⇒ |a + b| ≤ 2
@@ -1024,9 +1024,9 @@ a²+b²=1 ⇒ |a + b| ≤ 2 Which is what we wanted to prove, therefor the im

-
-
Reasoning by the Absurd:
-
+
+
Reasoning by the Absurd:
+

To prove that a proposition is True, we suppose that it’s False and we must come to a contradiction
@@ -1037,8 +1037,8 @@ And to prove that an implication P ⇒ Q is true using the reasoning by the absu

-
Example:
-
+
Example:
+

Prove that this proposition is correct using the reasoning by the absurd : ∀x ∈ ℝ* , sqrt(1+x²) ≠ 1 + x²/2
@@ -1056,17 +1056,17 @@ sqrt(1+x²) = 1 + x²/2 ; 1 + x² = (1+x²/2)² ; 1 + x² = 1 + x^4/4 + x² ;

-
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Reasoning by contraposition:
-
+
+
Reasoning by contraposition:
+

If an implication P ⇒ Q is too hard to prove, we just have to prove not(Q) ⇒ not(P) is true !!! or in other words that both not(P) and not(Q) are true

-
-
Reasoning by counter example:
-
+
+
Reasoning by counter example:
+

To prove that a proposition ∀x ∈ E, P(x) is false, all we have to do is find a single value of x from E such as not(P(x)) is true
@@ -1074,20 +1074,20 @@ To prove that a proposition ∀x ∈ E, P(x) is false, all we have to do is find

-
-
3eme Cours : Oct 9
-
+
+
3eme Cours : Oct 9
+

-
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Reasoning by recurrence :
-
+
+
Reasoning by recurrence :
+

P is a propriety dependent of n ∈ ℕ. If for n0 ∈ ℕ P(n0) is true, and if for n ≥ n0 (P(n) ⇒ P(n+1)) is true. Then P(n) is true for n ≥ n0

-
Example:
-
+
Example:
+

Let’s prove that ∀ n ≥ 1 , (n,k=1)Σk = [n(n+1)]/2
@@ -1123,21 +1123,21 @@ For n ≥ 1. We assume that P(n) is true, OR : (n, k=1)Σk = n(n+1)/2. We

-
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4eme Cours : Chapitre 2 : Sets and Operations
-
+
+
4eme Cours : Chapitre 2 : Sets and Operations
+

-
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Definition of a set :
-
+
+
Definition of a set :
+

A set is a collection of objects that share the sane propriety

-
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Belonging, inclusion, and equality :
-
+
+
Belonging, inclusion, and equality :
+

Let E be a set. If x is an element of E, we say that x belongs to E we write x ∈ E, and if it doesn’t, we write x ∉ E

A set E is included in a set F if all elements of E are elements of F and we write E ⊂ F ⇔ (∀x , x ∈ E ⇒ x ∈ F). We say that E is a subset of F, or a part of F. The negation of this propriety is : E ⊄ F ⇔ ∃x , x ∈ E and x ⊄ F
@@ -1146,13 +1146,13 @@ A set is a collection of objects that share the sane propriety

-
-
Intersections and reunions :
-
+
+
Intersections and reunions :
+

-
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Intersection:
-
+
+
Intersection:
+

E ∩ F = {x / x ∈ E AND x ∈ F} ; x ∈ E ∩ F ⇔ x ∈ F AND x ∈ F
@@ -1163,9 +1163,9 @@ x ∉ E ∩ F ⇔ x ∉ E OR x ∉ F

-
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Union:
-
+
+
Union:
+

E ∪ F = {x / x ∈ E OR x ∈ F} ; x ∈ E ∪ F ⇔ x ∈ F OR x ∈ F
@@ -1176,17 +1176,17 @@ x ∉ E ∪ F ⇔ x ∉ E AND x ∉ F

-
-
Difference between two sets:
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+
+
Difference between two sets:
+

E\F(Which is also written as : E - F) = {x / x ∈ E and x ∉ F}

-
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Complimentary set:
-
+
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Complimentary set:
+

If F ⊂ E. E - F is the complimentary of F in E.
@@ -1197,52 +1197,52 @@ FCE = {x /x ∈ E AND x ∉ F} ONLY WHEN F IS A SUBSET OF E

-
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Symentrical difference
-
+
+
Symentrical difference
+

E Δ F = (E - F) ∪ (F - E) ; = (E ∪ F) - (E ∩ F)

-
-
Proprieties :
-
+
+
Proprieties :
+

Let E,F and G be 3 sets. We have :

-
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Commutativity:
-
+
+
Commutativity:
+

E ∩ F = F ∩ E E ∪ F = F ∪ E

-
-
Associativity:
-
+
+
Associativity:
+

E ∩ (F ∩ G) = (E ∩ F) ∩ G E ∪ (F ∪ G) = (E ∪ F) ∪ G

-
-
Distributivity:
-
+
+
Distributivity:
+

E ∩ (F ∪ G) = (E ∩ F) ∪ (E ∩ G) E ∪ (F ∩ G) = (E ∪ F) ∩ (E ∪ G)

-
-
Lois de Morgan:
-
+
+
Lois de Morgan:
+

If E ⊂ G and F ⊂ G ;
@@ -1252,33 +1252,33 @@ If E ⊂ G and F ⊂ G ;

-
-
An other one:
-
+
+
An other one:
+

E - (F ∩ G) = (E-F) ∪ (E-G) ; E - (F ∪ G) = (E-F) ∩ (E-G)

-
-
An other one:
-
+
+
An other one:
+

E ∩ ∅ = ∅ ; E ∪ ∅ = E

-
-
And an other one:
-
+
+
And an other one:
+

E ∩ (F Δ G) = (E ∩ F) Δ (E ∩ G)

-
-
And the last one:
-
+
+
And the last one:
+

E Δ ∅ = E ; E Δ E = ∅
@@ -1289,7 +1289,7 @@ E Δ ∅ = E ; E Δ E = ∅

Author: Crystal
-
Created: 2023-10-11 Wed 19:04
+
Created: 2023-10-13 Fri 16:58

\ No newline at end of file -- cgit 1.4.1-2-gfad0

Crystal’s Website 💜

Welcome to Crystal’s Cozy Nook!

Welcome to Crystal’s Cozy Nook!

Articles ( NEW !!!! )

Articles ( NEW !!!! )

root@localhost $ whoami

root@localhost $ whoami

Sign my Guestbook (External website warning)

Sign my Guestbook (External website warning)

Blinkies

Blinkies

My banner

My banner

Webrings & Links (JAVASCRIPT WARNING)!!

Webrings & Links (JAVASCRIPT WARNING)!!

Misc :

Misc :

Algebra 1

Contenu de la Matiére

Contenu de la Matiére

Rappels et compléments (11H)

Rappels et compléments (11H)

Structures Algébriques (11H)

Structures Algébriques (11H)

Polynômes et fractions rationnelles

Polynômes et fractions rationnelles

Premier cours : Logique mathématique et méthodes du raisonnement mathématique Sep 25 :

Premier cours : Logique mathématique et méthodes du raisonnement mathématique Sep 25 :

Properties:

Properties:

Absorption:

Absorption:

Commutativity:

Commutativity:

Associativity:

Associativity:

Distributivity:

Distributivity:

Neutral element:

Neutral element:

Negation of a conjunction & a disjunction:

Negation of a conjunction & a disjunction:

Transitivity:

Transitivity:

Contraposition:

Contraposition:

God only knows what this property is called:

God only knows what this property is called:

Some exercices I found online :

Some exercices I found online :

USTHB 2022/2023 Section B :

USTHB 2022/2023 Section B :

2éme cours Oct 2

2éme cours Oct 2

Quantifiers

Quantifiers

Proprieties

Proprieties

Multi-parameter proprieties :

Multi-parameter proprieties :

Methods of mathematical reasoning :

Methods of mathematical reasoning :

Direct reasoning :

Direct reasoning :

Reasoning by the Absurd:

Reasoning by the Absurd:

Reasoning by contraposition:

Reasoning by contraposition:

Reasoning by counter example:

Reasoning by counter example:

3eme Cours : Oct 9

3eme Cours : Oct 9

Reasoning by recurrence :

Reasoning by recurrence :

4eme Cours : Chapitre 2 : Sets and Operations

4eme Cours : Chapitre 2 : Sets and Operations

Definition of a set :

Definition of a set :

Belonging, inclusion, and equality :

Belonging, inclusion, and equality :