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By Smith D., Eggen M., Andre R.

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4 14. Riddle: What is the English translation of the symbolic statement ∀E E ∀? Basic Proof Methods I In mathematics, a theorem is a statement that describes a pattern or relationship among quantities or structures and a proof is a justification of the truth of a theorem. Before beginning to examine valid proof techniques it is recommended that you review the comments about proofs and the definitions in the Preface to the Student. We cannot define all terms nor prove all statements from previous ones.

2. 3. 4. Determine precisely the hypotheses (if any) and the antecedent and consequent. Replace (if necessary) the antecedent with a more usable equivalent. Replace (if necessary) the consequent by something equivalent and more readily shown. Beginning with the assumption of the antecedent, develop a chain of statements that leads to the consequent. Each statement in the chain must be deducible from its predecessors or other known results. As you write a proof, be sure it is not just a string of symbols.

Qxd 14 CHAPTER 1 4/22/10 1:42 AM Page 14 Logic and Proofs This sentence clearly has the form of a conditional sentence, although almost everyone will interpret the meaning as a biconditional. Contrast this with the situation in mathematics where “If x = 2, then x is a solution to x 2 = 2x” must have only the meaning of the connective ⇒, because x 2 = 2x does not imply x = 2. Shown below are some phrases in English that are ordinarily translated by ⇒. In the accompanying examples, think of a and t as using the connectives ⇒ or ⇐ fixed real numbers.

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