Prove the following using the method suggested:
(a) Prove the following either by direct proof or bycontraposition:
Let a ? Z, if a ? 3 (mod 5) and b ? 2 (mod 5), then ab ? 1 (mod5).
(b) Prove the following by contradiction:
Suppose a, b ? Z. If a² + b²
is odd, then (2|a) ? (2|b), where ? is the exclusivedisjuntion,
i.e. p ? q = (p ? q) ? ¬(p ? q).
(d) Prove the following by cases: For all n ? Z,n
2 + 3(n + 1) is odd.
(e) Prove the following by induction:
For n ? 1,
1 × 2 + 2 × 3 + 3 × 4 + · · · + n(n + 1) = n/3
(n + 1)(n + 2)
