What are different methods of proof? Example with example.

A proof is a logical argument that tries to show that a statement is true. In math, and computer science, a proof has to be well thought out and tested before being accepted. But even then, a proof can be discovered to have been wrong. There are many different ways to go about proving something, we’ll discuss 3 methods: direct proof, proof by contradiction, proof by induction. 

Types of Proofs.

 Suppose we wish to prove an implication p → q. Here are some strategies we have available to try.

 • Trivial Proof: If we know q is true then p → q is true regardless of the truth value of p.

 • Vacuous Proof: If p is a conjunction of other hypotheses and we know one or more of these hypotheses is false, then p is false and so p → q is vacuously true regardless of the truth value of q.

 • Direct Proof: Assume p, and then use the rules of inference, axioms, definitions, and logical equivalences to prove q.

 • Indirect Proof or Proof by Contradiction: Assume p and ¬q and derive a contradiction r ∧ ¬r.

• Proof by Contrapositive: (Special case of Proof by Contradiction.) Give a direct proof of ¬q → ¬p. Assume ¬q and then use the rules of inference, axioms, definitions, and logical equivalences to prove ¬p.(Can be thought of as a proof by contradiction in which you assume p and ¬q and arrive at the contradiction p ∧ ¬p.)

 • Proof by Cases: If the hypothesis p can be separated into cases p1 ∨ p2 ∨ · · · ∨ pk, prove each of the propositions, p1 → q, p2 → q, . . . , pk → q, separately