  An MSTD set is a finite set with more pairwise sums than differences.
$(\Upsilon,\Phi)$-ismorphisms are generalizations of Freiman isomorphisms to
arbitrary linear forms. These generalized isomorphisms are used to prove that
every finite set of real numbers is Freiman isomorphic to a finite set of
integers. This implies that there exists no MSTD set $A$ of real numbers with
$|A| \leq 7$, and, up to Freiman isomorphism, there exists exactly one MSTD set
$A$ of real numbers with $|A| = 8$.
