In addition to the other answers, I'd like to add that the Schengen area, the EU, and the Eurozone are all technically separate, none is a subset of one of the others:
Ireland and Cyprus are in EU & Eurozone but not Schengen; Poland, Hungary (and more) are in EU & Schengen but not Eurozone; Switzerland is in Schengen but neither EU nor Eurozone; Montenegro and Kosovo are in the Eurozone but neither the EU nor Schengen.