Are prime numbers a set closed under addition?

While answering questions on , I ran into a lot of them
that talked about a set being 'closed'.  I didn't know what that
means, so do ya know what I did ?  I went and looked it up. 
I found it, and then I read about it, and now I know what it means !

A set that's 'closed under addition' means that if you add two numbers
in the set, you get another number that's also in the set.

Let's see:

-- 3 is a prime number
-- 5 is a prime number
-- (3+5) = 8
-- 8 is NOT a prime number

-- We added two prime numbers and got a number that is NOT prime.
   So the set of prime numbers is NOT closed under addition.