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 (P ar l’absurde / By contradiction) is a clever mathematical logic :**

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

**Example: Prove ** … (*)

Proof: (by reductio ad absurdum)

Assume the opposite of (*) is true:

[**Rigor**: Square both sides, “**<“** relation still kept since both sides are **positive** and **Square** is a * strictly monotonous (increasing) *function]

…

Hence, (*) is True :

The young teacher showed the techniques of proving Mapping (映射):

**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**:

To prove Surjective:

…

…

He used an analogy of (the Set of) red Indians shooting (the Set of) bisons 野牛:

ALLbisons are shot by arrows fromIndians. (1 or moreSurjectiveshoot)

**II.) Injective** (1-to-1) 单射

**By definition**:

To prove Injective, more practical to prove by contradiction:

…

…

prove: x = x’

**III.) Bijective** (On-to & 1-to-1) 双射

**Definition**:

To prove Bijective,

**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.”