Cours Raisonnements (Logics) , Ensembles ( Sets), Applications (Mappings)

This is an excellent quick revision of the French Baccalaureat Math during the first month of French university. (Unfortunately common A-level Math syllabus lacks such rigourous Math foundation.)

Most non-rigourous high-school students / teachers abuse the use of :

“=> ” , “<=>” . 

Prove by “Reductio ad Absurdum” 反证法 (Par l’absurde / By contradiction) is a clever mathematical logic :

\boxed {(A => B) <=> (non B => non A)}

Famous Examples: 1) Prove \sqrt 2 is irrational ; 2) There are infinite prime numbers  (by Greek mathematician Euclid 3,000 years ago).

Example: Prove \forall n \in {\mathbb{N}}^{*}, \frac {2n+1}{2 \sqrt {n(n+1)} } \geq 1 … (*)

Proof: (by reductio ad absurdum)
Assume the opposite of (*) is true:
\forall n \in {\mathbb{N}}^{*}, \frac {2n+1}{2 \sqrt {n(n+1)} } < 1
\iff {2n+1} < 2 \sqrt {n(n+1)}
\iff (2n+1) ^{2} < 4.n(n+1)
[Rigor: Square both sides, “<“ relation still kept since both sides are positive and Square is a strictly monotonous (increasing) function]

\iff 1 < 0  ,  \text {(False!) }
Hence, (*) is True : \boxed {\forall n \in {\mathbb{N}}^{*},\frac {2n+1}{2 \sqrt {n(n+1)} } \geq 1}

The young teacher showed the techniques of proving Mapping (映射):
E \to F
x \mapsto y = f (x)

Caution: A mapping from E to F always has ONE and ONLY ONE image in F.

I.) Surjective (On-to) –  best understood in Chinese 满射 (Full Mapping).
By definition:
\boxed { \forall y \in F, \exists x \in E, f (x) = y}

To prove Surjective:
\text {Let } y \in F


\text {find } \exists x \in E, f (x) = y 

He used an analogy of (the Set of)  red Indians shooting (the Set of) bisons 野牛: ALL bisons are shot by arrows from 1 or more Indians. (Surjective shoot)

II.) Injective (1-to-1) 单射

By definition: \boxed { \forall (x,x') \in E^{2}, x \neq x' \implies f (x) \neq f (x') }

To prove Injective, more practical to prove by contradiction:

\forall (x,x') \in E^{2}, \text { Suppose: } f (x) = f (x')


prove: x = x’

III.) Bijective (On-to  & 1-to-1) 双射
Definition:
\boxed{\forall y \in F, \exists ! x\in E, f (x) = y }

To prove Bijective,
\text {Let } y\in F, \text {let } x\in E, f(x) = y
\iff \text {...}
\iff \text {...}
\iff x = g (y)
\iff \boxed { g = f^{-1}}

My example: Membership cards are issued to ALL  club members (Surjective or On-to), and every member has one unique membership card identity number (1-to-1 or Injective), thus

“Cards – Members” mapping is Bijective.

(My Remark): If the mappings f and g are both surjective (满射), then

the composed mapping f(g) is also 满 (满) 射 = 满射 surjective ! (Trivial). [#]

He highlighted other methods of proof by higher math (Linear Algebra or Isomorphism).

Note [#]: “Abstract” Math concepts expressed in rich Chinese characters are more intuitive than the esoteric “anglo-franco-greco-germanic” terminologies. Some good examples are: homo-/endo-/iso-/auto-/homeo-morphism  (同态/自同态/同构/自同构/同胚), homology  (同调), homotopy (同伦), matrix (矩阵), determinant (行列式), eigen-value/vector (特征 值 /向量), manifold  (流形), simplicial (单纯) complex (复形), ideal (理想), topology  (拓扑), monad (单子), monoid (么半群)…

No wonder André Weil (WW2 Modern Math French/USA “Bourbaki School” Founder) had remarked:

“One day the westerners will have to learn Math in Chinese.”

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Mathematics: The Next Generation

Historical Backgroud:

Math evolves since antiquity, from Babylon, Egypt 5,000 years ago, through Greek, China, India 3,000 years ago, then the Arabs in the 10th century taught the Renaissance Europeans the Hindu-Arabic numerals and Algebra, Math progressed at a condensed rapid pace ever since: complex numbers to solve cubic equations in 16th century Italy, followed by the 17 CE French Cartersian Analytical Geometry, Fermat’s Number Theory,…, finally by the 19 CE to solve quintic equations of degree 5 and above, a new type of Abstract Math was created by a French genius 19-year-old Evariste Galois in “Group Theory”. The “Modern Math” was born since, it quickly develops into over 4,000 sub-branches of Math, but the origin of Math is still the same eternal truth.

Math Education Flaw: 本末倒置 Put the cart before the horse.

Math has been taught wrongly since young, either is boring, or scary, or mechanically (calculating).

This lecture by Queen Mary College (U. London) Prof Cameron is one of the rare Mathematician changing that pedagogy. Math is a “Universal Language of Truths” with unambiguous, logical syntax which transcends over eternity.

I like the brilliant idea of making the rigorous Math foundation compulsory for all S.T.E.M. (Science, Technology, Engineering, Math) undergraduate students. Prof S.S. Chern 陈省身 (Wolf Prize) after retirement in Nankai University (南开大学, 天津, China) also made basic “Abstract Algebra” course compulsory for all Chinese S.T.E.M. undergraduates in 2000s.

The foundations Prof Cameron teaches are centered around 4 Math Objects:

1. SET 集合
– Set is the founding block of the 20th century Modern Math, hitherto introduced into the world’s university textbooks by the French “Bourbaki” school (André Weil et al) after WW1.

Note: The last “Bourbaki” grand master Grothendieck proposed to replace Set by Category. That will be the next century Math for future Artificial Intelligence Era, aka “The 4th Human Revolution”.

2. FUNCTION 函数
– A vision first proposed by the German Gottingen School’s greatest Math Educator Felix Klein, who said Functions can be visualised in graphs, so it is the best tool to learn mathematical abstractness.

3. NUMBERS
– The German mathematician Leopold Kronecker, who once wrote that “God made the integers; all else is the work of man.”

– The universe is composed of numbers in “NZQRC” (ie Natural numbers, Integers, Rationals, Reals, Complex numbers). After C (Complex), no more further split of new numbers. Why?

4. Proofs
– reading and debugging proofs.

Example 1: Proof by Contradiction, aka Reductio ad Absurdum (Euclid’s Proof on Infinitely Many Prime Numbers)

image

Challenge the proof: Why ?
image
Induction intuitively by:
image

image

Example 2: Proof by Logic
image

[Hint:]
By Reasoning (which is unconscious), most would get “2 & A” (wrong answer)

By Logic (using consciousness), then you can proof …
Correct Answer: 2 and B
Test on all 3 Truth cases below in Truth Table:
p = front side
q = back side

image

Recommended The Best Book on Abstract Algebra by Prof. Peter J. Cameron

Math Education Evolution: From Function to Set to Category

Interesting Math education evolves since 19th century.

“Elementary Math from An Advanced Standpoint” (3 volumes) was proposed by German Göttingen School Felix Klein (19th century) :
1)  Math teaching based on Function (graph) which is visible to students. This has influenced  all Secondary school Math worldwide.

2) Geometry = Group

After WW1, French felt being  behind the German school, the “Bourbaki” Ecole Normale Supérieure students rewrote all Math teachings – aka “Abstract Math” – based on the structure “Set” as the foundation to build further algebraic structures (group, ring, field, vector space…) and all Math.

After WW2, the American prof MacLane & Eilenburg summarised all these Bourbaki structures into one super-structure: “Category” (范畴) with a “morphism” (aka ‘relation’) between them.

Grothendieck proposed rewriting the Bourbaki Abstract Math from ‘Set’ to ‘Category’, but was rejected by the jealous Bourbaki founder Andre Weil.

Category is still a graduate syllabus Math,  also called “Abstract Nonsense”! It is very useful in IT  Functional Programming for “Artificial Intelligence” – the next revolution in “Our Human Brain” !

Set Theory in John 1:1

“In the beginning was the Word, and the Word was with God, and the Word was God. ” (John 1:1)

太初有道。道与神同在,道就是神。

Mathematically in Set Theory:
Let W = Word, G = God

∃ W, (In the beginning was the Word)
and
W⊂ G, W⊃ G (the Word was with God)

=> W=G (the Word was God)

(John 14:11 NIV)
“Believe me when I say that I am in the Father and the Father is in me; or at least believe on the evidence of the works themselves.”

Proof: by Set Theory
⊂: in, include, inside

Jesus ⊂ Father (I am in the Father )
Father ⊂ Jesus (the Father is in me)
=> Jesus = Father (= God)
[QED]