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). [#]

On the way to the West, the team came to a Y-junction, either go Left or Right.
The Monkey King (孙悟空 ) went out reconnaissance, only came back few days later.
The Pig (猪八戒) suspects this fellow is a monster in disguise of the monkey, may not tell the truth.
The smart Monk (三藏) asks only 2 questions to detect his identity :

Q1. Is ‘Left’ direction correct ?
Q2. If I ask you Q1, you would say “Yes”, would-you?

The true Monkey will answer “Yes/Yes” or “No/No”, regardless the correct direction.
Why ?

1. We can choose to believe God exists, or we can choose not to so believe.
2. If we reject God and act accordingly, we risk everlasting agony and torment if He does exist (Type I error in Statistics lingo) but enjoy fleeting earthly delights if He doesn’t exist.
3. If we accept God and act accordingly, we risk little if He doesn’t exist (Type II error) but enjoy endless heavenly bliss if He does exist.
4. It’s in our self-interest to accept God’s existence.
5. Therefore God exists!

Mathematical Proof:
Pascal assumed
Probability of God exists = p
Probability God doesn’t exist = 1-p

You lead 2 lives, either Worldly (世俗) or Piously （虔,诚) , you get rewards X, Y, infinity or Z, as shown in table below.

In Worldly Life, the Expectation in probability is

Ew = p.X + (1-p).Y

In pious life, the Expectation is

Ep = p.∞ + (1-p).Z

Regardless how big the rewards in X (negative, punishment by God) ,Y, Z (self-sacrifice, negative reward)