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 :

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 Functional mapping:

Caution: A Function from E to F always has ONE and ONLYONE 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: ALL bisons are shot by arrows from 1 or more Indians. (Surjective shoot)

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). [#]