This question is quite tricky. To answer it right, you had to pay attention to its second part.

The thing is that the title of oldest public university in the United States is claimed by three universities: the University of Georgia, the University of North Carolina at Chapel Hill, and The College of William and Mary. Each has a distinct basis for the claim, North Carolina being the first to graduate a class, Georgia being the first created by charter, and William & Mary having the oldest founding date of any currently public university, though it was private for over 250 years.

The legislature chartered the University of North Carolina in 1789, and construction on the campus in Chapel Hill began in 1793. The University opened its doors in 1795 when it completed construction on its first building, Old East, and admitted its first students. UNC graduated its first class in 1798, becoming the first 18th century public institution in the US to confer degrees.

More Info: en.wikipedia.org