Does 'set' have an accepted definition in the context of multiple units? I've only ever used it as a synonym for coaching stock 'rakes'.
Off the top of my head, I'd have said that set is synonymous with multiple unit, and is a number of carriages in a semi-permanent configuration, bearing a set number. Sets are usually independently powered, either by electric (25kV ac / 750v dc / battery) and/or diesel engines, although some have been unpowered (eg 3/4 TC sets on the southern, and their derivatives, and more recently TPE's Mk5 formations).
Whereas I'd have said that a 'rake' is the vehicles comprising a whole train, and as such may comprise several sets, or a number of unpowered carriages, or, in times gone, by a mixture of sets and unpowered vehicles (eg dmu sets plus a parcels wagon or two).
Happy to be corrected!