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Diffstat (limited to 'src')
| -rwxr-xr-x | src/org/index.org | 22 | ||||
| -rwxr-xr-x | src/org/uni_notes/algebra1.org | 100 |
2 files changed, 101 insertions, 21 deletions
diff --git a/src/org/index.org b/src/org/index.org index a067e01..609b221 100755 --- a/src/org/index.org +++ b/src/org/index.org @@ -5,7 +5,7 @@ #+EXPORT_FILE_NAME: ../../index.html #+HTML_HEAD: <link rel="stylesheet" type="text/css" href="src/css/colors.css"/> #+HTML_HEAD: <link rel="stylesheet" type="text/css" href="src/css/style.css"/> -#+OPTIONS: \n:y + #+OPTIONS: html-style:nil #+OPTIONS: toc:nil * Welcome to Crystal's Cozy Nook! @@ -53,25 +53,7 @@ Want to leave a message, opinion, review or a salty insult ? Be sure to Sign my alt="Valid CSS!" /> </a> #+END_EXPORT -[[https://nishi.boats/][file:./src/gifs/blinkies/nishiboats.jpg]] -[[./src/gifs/blinkies/girlsnow.png]] -[[./src/gifs/blinkies/cookiefree.gif]] -[[./src/gifs/blinkies/transnow2.gif]] -[[./src/gifs/blinkies/gaywebring.gif]] -[[./src/gifs/blinkies/tranarchy.gif]] -[[./src/gifs/blinkies/button-torrents.gif]] -[[./src/gifs/blinkies/tyg.gif]] -[[./src/gifs/blinkies/fuck-google.gif]] -[[./src/gifs/blinkies/fuck_facebook.gif]] -[[./src/gifs/blinkies/graphics_by_gimp.gif]] -[[./src/gifs/blinkies/learn_html.gif]] -[[./src/gifs/blinkies/leave-twitter.gif]] -[[./src/gifs/blinkies/stop_microsoft.gif]] -[[./src/gifs/blinkies/web-pi.png]] -[[./src/gifs/blinkies/piracy.gif]] -[[./src/gifs/blinkies/best_viewed_with_eyes.gif]] -[[https://spyware.neocities.org/articles/discord][file:./src/gifs/blinkies/discord-no-way-2.gif]] -[[https://yesterweb.org/no-to-web3/][file:./src/gifs/blinkies/roly-saynotoweb3.gif]] +[[https://nishi.boats/][file:./src/gifs/blinkies/nishiboats.jpg]] [[./src/gifs/blinkies/girlsnow.png]] [[./src/gifs/blinkies/cookiefree.gif]] [[./src/gifs/blinkies/transnow2.gif]] [[./src/gifs/blinkies/gaywebring.gif]] [[./src/gifs/blinkies/tranarchy.gif]] [[./src/gifs/blinkies/button-torrents.gif]] [[./src/gifs/blinkies/tyg.gif]] [[./src/gifs/blinkies/fuck-google.gif]] [[./src/gifs/blinkies/fuck_facebook.gif]] [[./src/gifs/blinkies/graphics_by_gimp.gif]] [[./src/gifs/blinkies/learn_html.gif]] [[./src/gifs/blinkies/leave-twitter.gif]][[./src/gifs/blinkies/stop_microsoft.gif]] [[./src/gifs/blinkies/web-pi.png]] [[./src/gifs/blinkies/piracy.gif]] [[./src/gifs/blinkies/best_viewed_with_eyes.gif]] [[https://spyware.neocities.org/articles/discord][file:./src/gifs/blinkies/discord-no-way-2.gif]] [[https://yesterweb.org/no-to-web3/][file:./src/gifs/blinkies/roly-saynotoweb3.gif]] [[https://openbsd.org/][file:./src/gifs/blinkies/openbsd.png]] diff --git a/src/org/uni_notes/algebra1.org b/src/org/uni_notes/algebra1.org index 865d5b1..fa87d81 100755 --- a/src/org/uni_notes/algebra1.org +++ b/src/org/uni_notes/algebra1.org @@ -472,7 +472,7 @@ If F ⊂ E. E - F is the complimentary of F in E. FCE = {x /x ∈ E AND x ∉ F} *ONLY WHEN F IS A SUBSET OF E* -*** Symentrical difference +*** Symmetrical difference E Δ F = (E - F) ∪ (F - E) ; = (E ∪ F) - (E ∩ F) ** Proprieties : Let E,F and G be 3 sets. We have : @@ -594,3 +594,101 @@ De (1) et (2), P(0) est vraie ---- (a) Donc P(n+1) est vraie. ---- (b) De (a) et (b) on conclus que la proposition de départ est vraie #+END_VERSE +* Chapter 3 : Applications +** 3.1 Generalities about applications : +*** Definition : +Let E and F be two sets. +1. We call a function of the set E to the set F any relation from E to F such as for any element of E, we can find _at most one_ element of F that corresponds to it. +2. We call an application of the set E to the set F a relation from E to F such as for any element of E, we can find _one and only one_ element of F that corresponds to it. +3. f: E_{1} ---> F_{1} ; g: E_{2} ---> F_{2} ; f ≡ g ⇔ [E_{1 }= E_{2} ; F_{1} = F_{2} ; f(x) = g(x) ∀x ∈ E_{1} + + Generally speaking, we schematize a function or an application by this writing : + #+BEGIN_VERSE + f : E ---> F + x ---> f(x)=y + Γ = {(x , f(x))/ x ∈ E ; f(x) ∈ F} is the graph of f + #+END_VERSE +**** Some examples : +***** Ex1: +#+BEGIN_VERSE +f : ℝ ---> ℝ + x ---> f(x) = (x-1)/x +is a function, because 0 does NOT have a corresponding element using that relation. +#+END_VERSE +***** Ex2: +#+BEGIN_VERSE +f : ℝ^{*} ---> ℝ + x ---> f(x)= (x-1)/x +is, however, an application +#+END_VERSE +*** Restriction and prolongation of an application : +Let f : E -> F an application and E_{1} ⊂ E therefore : +#+BEGIN_VERSE +g : E_{1} -> F +g(x) = f(x) ∀x ∈ E_{1} + +g is called the *restriction* of f to E_{1}. And f is called the *prolongation* of g to E. +#+END_VERSE +**** Example +#+BEGIN_VERSE +f : ℝ ---> ℝ + x ---> f(x) = x^{2} + +g : [0 , +∞[ ---> ℝ + x ---> g(x) = x² + +g is called the *restriction* of f to ℝ^{+}. And f is called the *prolongation* of g to ℝ. +#+END_VERSE +*** Composition of applications : +Let E,F, and G be three sets, f: E -> F and g: F -> G are two applications. We define their composition, symbolized by g_{o}f as follow : + + +g_{o}f : E -> G . ∀x ∈ E (g_{o}f)_{(x)}= g(f(x)) +** 3.2 Injection, surjection and bijection : +Let f: E -> F be an application : +1. We say that f is injective if : ∀x,x' ∈ E : f(x) = f(x') ⇒ x = x' +2. We say that f is surjective if : ∀ y ∈ F , ∃ x ∈ E : y = f(x) +3. We say that if is bijective if it's both injective and surjective at the same time. +*** Proposition : +Let f : E -> F be an application. Therefore: +1. f is injective ⇔ y = f(x) has at most one solution. +2. f is surjective ⇔ y = f(x) has at least one solution. +3. f is bijective ⇔ y = f(x) has a single and unique solution. +** 3.3 Reciprocal applications : +*** Def : +Let f : E -> F a bijective application. So there exists an application named f^{-1} : F -> E such as : y = f(x) ⇔ x = f^{-1}(y) +*** Theorem : +Let f : E -> F be a bijective application. Therefore its reciprocal f^{-1} verifies : f^{-1}_{o}f=Id_{E }; f_{o}f^{-1}=Id_{F} Or : + + +Id_{E} : E -> E ; x -> Id_{E}(x) = x +*** Some proprieties : +1. (f^{-1})^{-1} = f +2. (g_{o}f)⁻¹ = f⁻¹_{o}g⁻¹ +3. The graphs of f and f⁻¹ are symmetrical to each other by the first bis-sectrice of the equation y = x +** 3.4 Direct Image and reciprocal Image : +*** Direct Image : + Let f: E-> F be an application and A ⊂ E. We call a direct image of A by f, and we symbolize as f(A) the subset of F defined by : + + +f(A) = {f(x)/ x ∈ A} ; = { y ∈ F ∃ x ∈ A y=f(x)} +**** Example : +#+BEGIN_VERSE +f: ℝ -> ℝ + x -> f(x) = x² +A = {0,4} +f(A) = {f(0), f(4)} = {0, 16} +#+END_VERSE +*** Reciprocal image : +Let f: E -> F be an application and B ⊂ F. We call the reciprocal image of E by F the subset f^{-1}(B) : + + +f^{-1}(B) = {x ∈ E/f(x) ∈ B} ; x ∈ f^{-1}(B) ⇔ f(x) ∈ B +**** Example : +#+BEGIN_VERSE +f: ℝ -> ℝ + x -> f(x) = x² +B = {1,9,4} +f^{-1}(B) = {1,-1,2,-2,3,-3} + = {x ∈ ℝ/x² ∈ {1,4,9}} +#+END_VERSE |