Sets and N-tuples

ℝ : The set of Real Number

ℕ : The set of Natural Numbers

ℕ= { 1 , 2, 3, ...}

ℤ: The set of Integers

ℤ = {..., -2, -1, 0, 1, 2, ...}

ℚ: The set of Rational Numbers

ℚ = {x ∈ ℝ | x = for same m,n ∈ ℤ}

Ǝ m,n ∈ ℤ, such that x=

= {x ∈ ℝ | x ≥ 0 }

For any set X, we can for n-tuples / lists of elements of X:

(1) = {x | x = (, ), ∈ ℝ}

x = (, )

(2, 2, 5) (2, 5, 2)

[Order matters, Repetition is ok]

(2) = { }, n = ℤ

N-Tuples vs. Sets:

🇺🇸 Usa= {R, W, B}

🇲🇫France = {R, B, W}

🇨🇳C = {R, Y}

No good!

🇨🇦 ≠ {R, W, R, R}

= {R, W} = {W, R}

[Order doesn't matter, no repetition]

S(x): "Statement about x is true"

P(x): "x has property P"

Universal Quantifier:

x∈ X: S(x) [For every x∈ X, S(x) true]

Existential Quntifier:

x∈ X: S(x) [There exists an x∈ X such that S(X) true]

x∈ X: S(x) [There exists a unique x∈ X such that S(X) true]

x ∈ ℝ: (True statement)

x ∈ ℝ: (False statement)

x ∈ ℕ: (True statement)

x ∈ ℝ: (True statement)

x ∈ ℝ: (False statement)

x ∈ ℕ: (True statement)

x ∈ ℤ: (False statement)

x,y ∈ ℝ: x+y = y+x (T) x,y ∈ ℝ: xy = yx (T) [commutative]

A,B ∈ Matrix : AB = BA (False, but there is counterexample)

Recall

Definition: For sets X and Y, the

Cartesian Product[데카르트 곱] of X and Y is

XY : ={(x,y) | x ∈ X, y ∈ Y}

For sets :

Definition: A function f: X—> Y for sets X and Y is a subset f X × Y that satisfies

∀x∈X: ∃! y∈Y : (x,y) ∈ f. we write f(x)=y for (x,y) ∈ f.

y = f(x) [ f(x) = y means x is in item y ]

X = {Usa, Canada, Mexico}

H = {N, S}

Let F: X—>Y be a function

The domain of f is dom f=X

The target of f is target f = Y

The image of a set S X under f is f(S): = {f(x)| x ∈ S} = {y∈Y | x∈S: f(x) = y}

The range f is f(x)

The pre image of a set V Y is (v): = { x ∈ X | f(x) ∈ V}

f is onto Y if ∀ y ∈ Y :∃ x ∈ X: f(x) = y

f is one-to-one if ∀ y ∈ f(x) :∃! x ∈ X, f(x) = y

f is a 1-to-1 correspondence between x and y if ∀ y ∈ f(x) :∃! x ∈ X, f(x) = y