Truth, Deduction, ComputationLecture HSet Theory, Informally

Vlad PatryshevSCU2013

Sets, Informally

1. a ∈ A : a is an element of A2. a ∉ A ≡ ¬(a∈A)3. empty set ∅ ≡ ¬∃x (x∈∅)4. A ⊂ B ≡ ∀x ((x∈A)→(x∈B))5. A ⊄ B ≡ ¬(A⊂B)6. {a,b,c…} : a set consisting of a,b,c…7. {x∈A | P(x)} : set comprehension (like in Python)8. A ∩ B = {x | (x∈A) ∧ (x∈B)}9. A ∪ B = {x | (x∈A) ∨ (x∈B)}

10. A \ B = {x | (x∈A) ∧ ¬(x∈B)}11. Do we have two monoids?12. powerset P(A) ≡ {B | B ⊂ A} e.g. P(∅)={∅}; P(P(∅)={∅,{∅}}

Examples of Sets

1. empty set ∅ - size 02. finite sets e.g. {∅, {∅, {∅}}, {{∅}}}3. set of natural numbers ℕ - size ℵ

04. set of integer numbers ℤ - size ℵ

05. set of rational numbers ℚ - size ℵ

06. set of real numbers ℝ - size ℵ

17. set of complex numbers ℂ - size ℵ

1

Sets Equality

● {a,b} = {b,a}● {a,a,a} = {a}● A = B iff {A} = {B} iff {{A}} = {{B}}

A=B ≡ ∀x ((x∈A) ↔ (x∈B))

Powerset

● 2∅ ={∅}● 2{a,b} = {∅,{a},{b},{a,b}}● 2A ∪ B = 2A × 2A

P(A) aka 2A ≡ {x|x⊂A}

Natural Numbers in Sets

∅, {∅}, {{∅}}, {{{∅}}}, etc…

Meaning, 0≡∅; S(n)≡{n}. Just count the curlies.

Or, better, ∅, {∅}, {∅, {∅}}, {∅, {∅}, {{∅}}}

Where’s Universal Property? Oops, something’s missing.But first, introduce pairs.

Define Pair

(a,b), for a and b of any nature.No, {a,b} won’t work, it’s the same as {b,a}

How about {a,{a,b}}?

Almost there… do you see the problem?

Will fix it later.

Have Pairs, define Relationships

● A × ∅ = ∅● A × (B∪C) = (A×B) ∪ (A×C)● {a,b} × {x,y} = {(a,x),(a,y),(b,x),(b,y)}

Do you see a monoid?

A × B ≡ {(a,b) | (a∈A)∧(b∈B)}

Kinds of Binary Relationships

● xRy ≡ (x,y)∈R● reflexive: ∀x (xRx)● symmetric: ∀x∀y (xRy → yRx)● antisymmetric: ∀x∀y ¬(xRy ∧ yRx)● transitive: ∀x∀y∀z ((xRy ∧ yRz) → xRz)● equivalence: reflexive, symmetric, transitive● partial order: antisymmetric, reflexive● functional: ∀x ∃!y (xRy)

R ⊂ A × B

Infinities

● finite: in bijection with a natural number ({∅, {∅}, {{∅}}...})

● ℵ0 - in bijection with ℕ

● ℵ1 - in bijection with 2ℕ, which is in bijection with ℝ

● ℵ2 = 2ℵ1

● etc, ℵk+1

= 2ℵk

● ℵx= ∪ {ℵ

y| y < x}

● ℵ = the power of all sets; ℵ=2ℵ (?!)

A set is infinite if it is in bijection with its proper subset

Good Reading on Aleph

Many Definitions, Many Paradoxes

● Are there as many natural numbers as their squares?● Points of a square vs points on the edge of the square -

same amount?!● The diary of Tristram Shandy problem● Ross-Littlewood-Achilles (take 1 ball, add 10, each

time twice faster)● Russell paradox● König's paradox (the first real number not finitely

defineable)● etc

That’s it for today