Math 320 { September 02, 2018 2 Proof Inductive reasoning is the type of reasoning in which one draws a conclusion about the general case from considering particular examples. Deductive reasoning is the type of reasoning in which one draws a conclusion by applying a general principle to a particular case. Inductive reasoning by itself does not constitute a proof. One needs to use a deductive argument to prove the conclusion, even if the conclusion was first obtained by inductive reasoning. Most theorems can be formulated in the form p)q, in which case p is called the hypothesis and q is called the conclusion. When constructing a proof of the implication p)q, one usually builds a logical bridge of simpler implications: p)p1 )p2···)···)q2 )q1 )q: The contrapositive of p)q is ∼q)∼p, and it is equivalent to the original implication, i.e. (p)q),(∼q)∼p): The converse of p)q is q)p, and it is logically independent from the original implication, so (p)q)<(q)p): The inverse of p)q is ∼p)∼q, and it is the contrapositive of the converse of p)q, so (∼p)∼q),(q)p)<(p)q): 2.1 Techniques of Proof A direct proof is a type of proof in which one constructs a direct logical bridge from the hypothesis to the conclusion. For statements involving the universal quantifier 8, e.g. 8x;p(x), a direct proof involves proving that the statement p(x) is true for an arbitrary x in the system in which the original statement was made. One can also disprove the statement 8x;p(x) by giving a counterexample, i.e. by finding an x for which ∼p(x) holds. For statements involving the existential quantifier 9, e.g. 9x;p(x), a direct proof involves constructing, guessing, or somehow else producing an x, for which p(x) is true. An indirect proof is a type of proof in which one proves an equivalent statement to the original implication instead of constructing a direct logical bridge from the hypothesis to the conclusion. Examples of indirect proofs are proof of the contrapositive and proof by contradiction. For the later, one relies on the tautologies (c denotes a contradiction - a statement which is always false): (∼p)c),p and [(p^∼q))c],(p)q): Another useful technique (that can be used both in a direct proof and an indirect argument) is the proof by cases. If the hypothesis can be broken into several cases, then it is enough to prove the conclusion for each case separately. This is given by the tautology [(p_q))r],[(p)r)^(q)r)]: If the conclusion can be broken into several cases, it is enough to prove one of the cases assuming all the other cases of the conclusion are false. This is given by the tautology [p)(q_r)],[(p^∼q))r]:.