Settings

• People mountain people sea

• P = {🧕,👩‍⚕️,👷,💂‍♀️,🕵️‍♂️,👨‍🌾,👩‍🍳,…}

• belongs to (∈)

  1. 👨‍🔬 ∈ P
  2. 🐢 ∉ P

What is function?

	need better words to describes things what about inverse functions?

Name list

Z	M	S	G
姓	名	surname	given name
王	八旦	Wong	Eight Eggs
李	崔牛	Li	Tsui Ngau
牛	柏葉	Ngau	Albert
丁	一叮	Ting	One
李	鹵味	Lee	Braised Dishes
李	老竇	Li	Lo Dull
黃	土地	Wong	Land
李	老表	Li	Low Bill
李	腦細	Lee	Old boss
袁	尼姑	Yuen	Nun
容	海龜	Yung	Hoi Kwai
李	老友	Lee	No Friend
李	老板	Lee	Low Ban
任	何人	Yam	Ho Yan
歐陽	李	Au Yeung	Lee
容	海歸	Yung	Hoi Kwai
楊	祖	Yeung	Joe
麥	麥宋	Mak	Mac Delivery
狄	狄尼	Dick	Disney

Two ways to write sets

Goal: To write a set of all persons with surname “李”

• List:

  L = {(李,崔牛), (李,鹵味), (李,老竇), (李,老表), (李,腦細), (李,老友), (李,老板)}

• Condition:

  L = {🏃‍♂️ ∈ P | 🏃‍♂️ with surname “李”}

• ordered pair: (a,b)

  (歐陽,李) ≠ (李,歐陽)

• subset (⊆/⊂): L ⊆ P

  L is contained in P.
  ℹ️ I prefer using ‘⊆’.

• superset (⊇/⊃): P ⊇ L

• empty set (∅): a set that has no element.

ETV

• {🐢,🦆,😸,🐷,😸,🐢,🦆} = {🐢,🦆,😸,🐷}

set operations

union (∪)

• L₁ = {🏃‍♂️ ∈ P | 🏃‍♂️ with surname “Li”}
• L₂ = {🏃‍♂️ ∈ P | 🏃‍♂️ with surname “Lee”}
• L₁ ∪ L₂ = {🏃‍♂️ ∈ P | 🏃‍♂️ ∈ L₁ or 🏃‍♂️ ∈ L₂} = L
• Observation: L₁ ⊆ L and L₂ ⊆ L
• Exercise: write another set union using the surname “Wong”

intersection (∩)

• J = {(袁,尼姑), (李,老師), (容,海歸), (李,老板), (李,腦細)}
• L = {🏃‍♂️ ∈ P | 🏃‍♂️ with surname “李”}
• J ∩ L = {🏃‍♂️ ∈ P | 🏃‍♂️ ∈ J and 🏃‍♂️ ∈ L} = {(李,老師), (李,老板), (李,腦細)}
• Observation: J ∩ L ⊆ J and J ∩ L ⊆ L

complement (^c)

• L^c = {🏃‍♂️ ∈ P | 🏃‍♂️ ∉ L} = {🏃‍♂️ ∈ P | 🏃‍♂️ whose surname is NOT “李”}

difference (\)

• J \ L = {🏃‍♂️ ∈ P | 🏃‍♂️ ∈ J and 🏃‍♂️ ∉ L} = {(袁,尼姑), (容,海歸)}
• L^c = P \ L
• Observation: J = (J ∩ L) ∪ (J \ L)

Cartesian product

Logical AND (∧)

Example: Save (Ctrl + S)

Ctrl	S	Ctrl + S
✓	✓	✓
✓	✗	✗
✗	✓	✗
✗	✗	✗

p	q	p ∧ q
T	T	T
T	F	F
F	T	F
F	F	F

Logical OR (∨)

Example: digit key (e.g. 1 in numpad or above alphabets)

numpad 1	alphabet 1	enter 1
✓	✓	✓
✓	✗	✓
✗	✓	✓
✗	✗	✗

p	q	p ∨ q
T	T	T
T	F	T
F	T	T
F	F	F

Implication (⟹)

Example 1

entered Univ through JUPAS	attended secondary school	entered Univ through JUPAS ⟹ attended secondary school	example
✓	✓	✓	me
✓	✗	✗	who?
✗	✓	✓	IB, GCE A Level
✗	✗	✓	沈詩鈞

Implication (⟹)

Example 2

Promise: win Mark Six (六合彩) ⟹ treat you to dinner in a seafood restaurant

P	bought Mark Six?	bill on me?	promise kept?
👩‍🍳	✓	✓	✓
👮	✓	✗	✗
👨‍🌾	✗	✓	✓
👨‍💻	✗	✗	✓

👨‍💻: I don’t buy Mark Six, so I don’t have to spring for your dinner. 😛

set of persons who kept their promise = {🏃‍♂️ ∈ P | 🏃‍♂️ kept his/her promise} = {👩‍🍳, 👨‍🌾, 👨‍💻}

Implication (⟹) and subsets

p	q	p ⟹ q
T	T	T
T	F	F
F	T	T
F	F	T

Recall:

• L₁ = {🏃‍♂️ ∈ P | 🏃‍♂️ with surname “Li”}
• L = {🏃‍♂️ ∈ P | 🏃‍♂️ with surname “李”}
• Observation:
  • L₁ ⊆ L
  • 🏃‍♂️ ∈ L₁ ⟹ 🏃‍♂️ ∈ L
• Exercise: show that for any set A, ∅ ⊆ A.

Implication (⟹)

To prove a “⟹” statement

Direct way

1. Assume the premise p is true.
2. Show that the conclusion q is true.

Implication (⟹)

To prove a “⟹” statement

By contradiction

Example: solve Sudoku (數獨) by guessing.

1. p True
2. p ∧ ~q False
3. so ~q False (i.e. q True)

Sudoku source

Wiki’s Function definition

• ∀: for all
• ∃: there exists

A function f: A → B is a subset of A × B such that

The domain of f is A.

Domain, codomain and range

In the picture,

• the domain of f is A = {👷,👨‍🌾,👨‍💻,👨‍💼,👨‍⚖️}
• the codomain of f is B = {🧕,👮‍♀️,👩‍🎤,👩‍🏫,👩‍✈️,👸}
• the range of f is f[A] = {🧕,👩‍🎤,👩‍🏫,👩‍✈️}.

Injective functions and Surjective functions

Bijective functions and inverse functions

A function is bijective is it is both injective and surjective.

1. surjective function (冇單身狗乸)

   Given 👩 ∈ B. You can find a 🧑 ∈ A so that 🧑❤👩.

2. injective function (女版用情專一)

   You can’t have two different 🧑 & 👦 such that 🧑❤👩 & 👦❤👩. They must been the same person. (represented by 🧑)

If we have a bijective function f : A → B defined by f : 🧑 ↦ 👩, then we can define an inverse function f⁻¹ : 👩 ↦ 🧑.

Comments and critiques

Field medalist Shing-Tung Yau (丘成桐) has emphasized the importance of asking good questions (not in the textbook).