Massachusetts Institute of Technology: Srini Devadas and Eric Lehman's "Proofs"
Read the opening discussion, "Proofs," Section 1, "The Axiomatic Method," and Sections 2 - 6, inclusive. In Unit 3, you will get a chance to apply a lot of what you have thought about and mastered in Units 1 and 2. Our domain of discourse is number theory, in particular proofs in elementary number theory.
Consider the following comparison of direct and indirect proofs. A direct proof is an argument that shows that the conclusion logically follows from the premises or assumptions by applying rules of inference in a sequence of steps.
Sections 1, 2, 4, and 5 deal with direct proofs. Section 2, "Proving an Implication," refers to statement such as P implies Q, and proving that Q is a consequence of P. In a logic system, P logically follows from the axioms and valid statements of the logic system, is another way of saying that P is a consequence of the axioms and valid statements or P is implied by them. This is what is meant by proving an implication.
Section 3 is related to indirect proof. Section 6 discusses indirect proof, also known as proof by contradiction, directly. An indirect proof, in contrast, is a proof in which the theorem to be proved is assumed false, and from this assumption, it is shown that a contradiction follows. Because the logic system is consistent, i.e. there are no contradictions, the theorem must be true. (Two statements are contradictions when they cannot both be true, and they cannot both be false.)