You're completely wrong.
When Hyundai first started selling cars in the US it's true they did use Mitsu engines, for example in the first-gen Elantra ('92-'95). Mitsu basically gave them their old engine casts and dies, which accounts for many of the quality problems Hyundai initially had. Mitsu basically gave them the parts that weren't much good anymore or had alignment problems etc.
Hyundai started developing it's own engines, and now uses mostly it's own. The 2.0L is called the Beta, as the first engine they developed was the Alpha. The 2.4L in the Sonata is a Theta and the 2.7L in the Tiburon is a Delta. The only engine that isn't completely Hyundai anymore is the so-called "World Engine" that it developed with Mitsu and Chrysler, which Mitsu uses in the new Evo, Chrysler uses in the Dodge Caliber, and I believe Hyundai will use in the upcoming Genesis coupe. Every other engine Hyundai has right now (3.3L, 3.8L, 4.6L V8, 1.6L etc) are proprietary Hyundai engines (IE: Hyundai developed them, nobody else).