Prove the following by contradiction and

Question

Prove the following using the method suggested:

(a) Prove the following either by direct proof or by contraposition:

Let a ∈ Z, if a ≡ 1 (mod 5), then a² ≡ 1 (mod 5).

(b) Prove the following by contradiction:

Suppose a, b ∈ Z. If 4|(a² + b²), then a and b are not both odd.

(c) Disprove the following by counter examples:

• For every natural number n, the integer n² + 17n + 17 is prime.
• Let A,B and C be sets. If A x C =  B x C, then A= B.

(d) Prove the following by cases: For all n ∈ Z, n² + 3n +4 is even.

(e) Prove the following by induction:

1² + 3² + 5² + ........ + (2n-1)² = n/3 (2n-1) (2n+1)