Logic & Math (Set Theory)

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: