(a) If A?C?B?Cthen A?B.
(b) If A?C?B?Cthen A?B.
4. [8 Pts] Prove that if A?C?B?Cand A?C?B?Cthen A?B. Hint: You can
either do a proof by contradiction or a proof by cases.
5. [8 Pts] Prove that (A?B) ?(A?B) = (A?B) ?(B?A) using subset argument.
You may not use logical equivalences in your proof. Use general proof techniques like
2oof by contradiction!nd 2oof by cases¼br>
6. [4 Pts] Prove that f(n) = 5n+ 9 is one-to-one, where the domain and co-domain of
fis Z+. Show that fis not onto.
7. [4 Pts] Prove that f(m,n) = m+ n+ mnis onto, where the domain of fis ZÚ
and the co-domain of fis Z. Show that fis not one-to-one.
8. [8 Pts] Let gbe a total function from Ato Band fbe a total function from Bto C.
(a) If f?gis one-to-one, then is gone-to-one? Prove or give a counter-example.
(b) If f?gis onto, then is gonto? Prove or give a counter-example.