Set union

In set theory, a union of two sets is a set containing all elements of both. To indicate unions, we use the symbol ∪, and say that if Γ and Δ are sets, then Γ ∪ Δ is a set containing all elements of Γ and Δ.

For example, suppose that Γ and Δ are sets of formulae, as defined as:

  • Γ = { α, β → γ, ¬γ }
  • Δ = { φ, δ ∨ α }

The union Γ ∪ Δ is therefore the following set:

  • Γ ∪ Δ = { α, β → γ, ¬γ, φ, δ ∨ α }

Set unions can also be defined by simply adding a single element to a set. For example, if we wanted to only add the formula φ to the set Γ, we would write:

  • Γ ∪ { φ }

The union of two sets can be defined as follows:

  • Γ ∪ Δ = {x| x∈Γ ∨ x∈Δ}

Which means, for any x that is an element of Γ ∪ Δ, x is an element of Γ, or x is a member of Δ (or both).