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).