# Primer: Set Theory

<https://en.wikipedia.org/wiki/Implementation_of_mathematics_in_set_theory>

* \[primitive] A set is a collection of objects
* set builder notation: { ∀x | Φ(x) }
* the empty set, `∅ ≝ { ∀x | x ≠ x }`
* singleton set,    `{x} ≝ { ∀y | y = x }`
* unordered pair, `{x,y} ≝ { ∀z | z = x ⋁ z = y }`
* union, `x ∪ y ≝ { ∀z | z ∈ x ⋁ z ∈ y}`
* ordered pair
  * K.Kuratowski: `(x,y) ≝ { {x}, {x,y} }`
  * N.Wiener:     `(x,y) ≝ { {{x},∅}, {{y}} }`
* recursive definition of unordered `n`-tuples for any concrete `n`&#x20;

  (finite sets given as lists of their elements):

  `{x₁, …, xₙ, xₙ﹢₁} ≝ {x₁, …, xₙ} ∪ {xₙ﹢₁}`
* ∀x. Ψ(x) → Φ(x)   where, e.g., Ψ(x) \~ x ∈ ℕ and Φ(x) \~ x >= 0
* ∃x. Ψ(x) ∧ Φ(x)
* Russell's paradox: { x | x ∉ x }
* the empty set
  * {} or ∅
  * ∅ ≝ { ∀x | x ≠ x }
  * E = { ∀x | x ∉ E } = { ¬∃x.x ∈ E }
* singleton set
  * For each object x, there is a set {x} with x as its only element
  * {x} ≝ { ∀y | y = x }
* unordered pair
  * For all objects x and y, there is a set {x,y}
  * {x,y} ≝ { ∀z | z = x ⋁ z = y }
* powerset:
  * 𝓟(A) = { ∀S | S ⊆ A }
* union:        A ∪ B = { ∀x | x ∈ A ∨ x ∈ B }
* intersection: A ∩ B = { ∀x | x ∈ A ∧ x ∈ B }
* difference:   A  B = { ∀x | x ∈ A → x ∉ B }
* difference:   B  A = { ∀x | x ∈ B → x ∉ A }
* symmetrical difference:
  * all items in A or B that are not in their intersection
  * A △ B = (A ∪ B)  (A ∩ B)
  * A △ B = (A  B) ∪ (B  A)
  * A △ B = { ∀x∀y | (x ∈ A → x ∉ B) ∨ (y ∈ B → y ∉ A) }

𝓤 = {a,b,c,d,e} S = {a,b,c}, T = {b,c,d}

* {a,b,c} ∪ {b,c,d} = {a,b,c,d}
* {a,b,c} ∩ {b,c,d} = {b,c}
* {a,b,c}  {b,c,d} = {a}
* {b,c,d}  {a,b,c} = {d}
* {a,b,c} △ {b,c,d} = {a,d}
* {a,b,c}' = {d,e}
* {b,c,d}' = {a,e}