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# section0notes - • Examples of disproof by counterexample...

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Lecture 0: Introduction to logic, and techniques of proof. 1. Truth tables and quantifiers Definition of a statement . Ex: 2 + 2 = 4. Non-ex: x 2 - 5 x + 6 = 0. Ex: For all x in the set { 0 , 1 , 2 , . . . } , x 2 - 5 x + 6 = 0. Truth tables for p , p q , and p q . A contradiction, p ∧ ∼ p , and a tautology p ∨ ∼ p . Negating p q , and p q . Ex: H is a normal subgroup of G . Ex: K is convex and bounded. Ex: I is open or closed. p q and logical equivalence. Work out truth table to see why p and q are equivalent if p q is true. Negating p q using p ∧ ∼ q . Show via truth table. Ex: If a sequence is Cauchy, then it’s convergent. Definitions of universal and existential quantifiers . Ex: For all x , x 2 - 5 x + 6 = 0. Ex: There exists an x so that x 2 - 5 x + 6 = 0. Negations of statements involving quantifiers. Ex: There exists a positive number x such that x 2 = 5. Ex: For every positive number M there exists a positive number N such that N < 1 /M . Ex: If n N , then | f n ( x ) - f ( x ) | ≤ 3 for all x in A . Ex: > 0 N so that n N , then x S , | f n ( x ) - f ( x ) | < . 2. Techniques of proof
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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 1-δ < 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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