a) Write one theory and prove it by using Proof by Contradiction,

The method of Proof by Contradiction. 1. Assume that P is true. 2. Assume that Q is true. 3. Use P and Q to demonstrate a contradiction.

b) Write one theory and prove it by using Proof by Induction

The method of Proof by Induction. 1. Show that a propositional form P(x) is true for some basis case. 2. Assume that P(n) is true for some n, and show that this implies that P(n+1) is true. 3. Then, by the principle of induction, the propositional form P(x) is true for all n greater or equal to the basis case.