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# Definitions

## Cartesian Product

If S and T are sets then the Cartesian product S×T of S and T is the set of ordered pairs (x,y) with x ∈ S and y ∈ T, denoted by S×T = { (x,y) | x ∈ S ∧ y ∈ T }

Alternatively: (x,y) ∈ S×T ⟺ ∀x∀y. x ∈ S, y ∈ T

More specifically: ∀p . p ∈ S×T ⟺ ∃x∃y. x ∈ S ∧ y ∈ T ∧ p = (x,y)

My: S×T = { (x,y) | ∀x∀y. x ∈ S ∧ y ∈ T }

> If S and T are sets, then the **Cartesian product** S×T of S and T is the set of all the ordered pairs (x,y) with x in S and y in T.
>
> In a Cartesian product S×T, the sets S and T are called the **factors** of S×T.
>
> The power set of a set S is the set defined and denoted as: P(S) = {∀A.A ⊆ S}. That is, the set whose elements are all of the subsets of S.

𝒰 = { x | ∀x. x ∈ 𝒰 } E = { ∀x ∈ 𝒰. ¬∃x ∈ E } = ∅ S ⋃ T = { ∀x. x ∈ S ∨ x ∈ T } S ⋂ T = { ∀x. x ∈ S ∧ x ∈ T } S  T = { ∀x. x ∈ S ∧ x ∉ T } S' = { x | ∀x. x ∈ 𝒰 ∧ x ∉ S } = 𝒰 ⋂ S

∀S. 𝒰 ⊇ S ∀S. 𝒰 ⋃ S = 𝒰 ∀S. 𝒰 ⋂ S = S' ∀S. ∅ ⊆ S ∀S. ∅ ⋃ S = S ∀S. ∅ ⋂ S = ∅

𝓟(𝒰) = ∀S. S is a set 𝓟(S) = { ∅, ..., {S} }

𝓟(∅) = {∅} The empty set ∅ has no elements but it does have one subset, ∅.