Extended set operations

Union and itersection of two sets (where 𝓤 is the universal set):

• Union

  • A₁ ∪ A₂ = { x ∈ 𝓤. x ∈ A₁ ∨ x ∈ A₂ }

• Union via the set index:

  • A₁ ∪ A₂ = { x ∈ 𝓤. ∃i ∈ {1,2}. x ∈ Aᵢ }

• Itersection

  • A₁ ∩ A₂ = { x ∈ 𝓤. x ∈ A₁ ∧ x ∈ A₂ }

• Itersection via the set index:

  • A₁ ∩ A₂ = { x ∈ 𝓤. ∀i ∈ {1,2}. x ∈ Aᵢ }

So, the correspondance is:

• ∪ : ∨ : ∃i (there is an index i)

• ∩ : ∧ : ∀i (for every index i)

Union/itersection of more then two sets use the big-ass union/itersection sign.

• (i=1..n) ⋃Aᵢ = { x ∈ 𝓤. ∃i ∈ {1..n}. x ∈ Aᵢ }

• (i=1..n) ⋂Aᵢ = { x ∈ 𝓤. ∀i ∈ {1..n}. x ∈ Aᵢ }

where n ∈ ℕ

 ₙ
 ⋃ Aᵢ = ⁿ⋃ᵢ₌₁ Aᵢ = ᵢ₌₁⋃ⁿ Aᵢ
ⁱ⁼¹
 ₙ
 ⋂ Aᵢ = ⁿ⋂ᵢ₌₁ Aᵢ = ᵢ₌₁⋂ⁿ Aᵢ
ⁱ⁼¹