# Saying either or both in first-order predicate logic?

I have a statement along the gist of "Each member of A is either Y, Z, or both"

Would the appropriate way to write this using first-order logic be:

∀x (A(x) → (Y(x) ∨ Z(x)))

Would it suffice to just use a disjunction here? Or would this be written as

∀x (A(x) → (Y(x) ∨ Z(x)) ∨ (Y(x) ∧ Z(x)))

or is this redundant?

• don't you know what a disjunction exactly is ? Oct 13, 2020 at 17:33

Since $$(A \land B) \implies (A \lor B)$$ the second clause in $$(Y(x) \lor Z(x)) \lor (Y(x) \land Z(x))$$ is redundant.