Math 117 – March 31, 2011 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 ⇒ p 1 ⇒ p 2 ··· ⇒ ··· ⇒ q 2 ⇒ q 1 ⇒ q. The contrapositive of p ⇒ q is ∼ q ⇒∼ p , and it is equivalent to the original implication, i.e. ( p
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