No, you need antivirus. Spybot is an excellent little program.
AVGfree is free

highly recommended.
Norton is crap, I'm speaking from personal experience.
I had norton antivirus and firwall on another computer and for a long time (months) liveupdate would'nt work because a trojan got in. After a search on the internet i eventually found out others had the same problem and norton would'nt could'nt fix it so i went to PCTOOLS.COM
Which i would highly recommend if you dont mind paying for your software.