 This fact allows us to say the free algebra on a given set, thinking of isomorphic algebras as being “morally” the same. Were this not the case, our quotient construction would be incomplete as it produces a unique free algebra, whereas the above definition of free algebra allows any algebra isomorphic to that produced by the quotient construction to be considered free. Since all free algebras on \(X\) are isomorphic, the quotient construction is as good as any, and is furthermore one way of proving that they exist. It also establishes that the choice of set of variables is irrelevant except for its cardinality, as intuition would suggest.