Unformatted text preview: • Examples of disproof by counterexample. Ex: For every real number x , if x 2 > 4 then x > 2. • Examples of proof by contrapositive and contradiction. Show the contrapositive ∼ q ⇒∼ p is equivalent to p ⇒ q . Ex: Use the contrapositive to prove “If 7 m is an odd number, then m is an odd number.” Proof by contradiction: Assume p true and q false. Contradict a known theorem (or axiom). (Note the ﬁne line between contradiction and contrapositive). Ex: Let x be a real number. If x > 0 then 1 /x > 0. • Examples of the logical structure of proofs involving quantiﬁers. Theorem: For every ± > 0, ∃ δ > 0 such that 1δ < x < 1 + δ implies that 5± < 2 x + 3 < 5 + ± . Theorem: Let f be a continuous function. If R 1 f ( x ) dx 6 = 0, then ∃ x ∈ [0 , 1] so that f ( x ) 6 = 0. 1...
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 Winter '11
 Wiley
 Differential Equations, Logic, Equations

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