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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