if a is bad, then negative bad or not bad is good.

-a means the opposite of a. Since a is negative and the opposite of a negative is positive, -a is positive.

The smallest positive prime is 2, but there is no smallest negative prime.

One way to show this is to demonstrate that the cardinality of the set of negative primes is countably infinite (the proof will be similar to that for positive primes. Hint: use proof by contradiction. Assume a finite set of negative primes and derive a contradiction).

Let me know if you'd like me to write up a formal proof that there is no smallest negative prime. I'll be happy to dig it out of one of my old homework sets! :)

a negative number plus a negative number is negative. here is a proof.

(-x)+(-x)

(-1)x+(-1)x

-1(x+x)

-1(2x)

-2x

a negative number times a negative number is positive though.