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:
The union Γ ∪ Δ is therefore the following set:
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:
The union of two sets can be defined as follows:
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).