# lecture09 - MA1100 Lecture 9 Mathematical Proofs Proof by...

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1 MA1100 Lecture 9 Mathematical Proofs Proof by Contradiction Existence Proof Chartrand: section 5.2, 5.4

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Lecture 9 2 Proof by Contradiction To prove statement R is true Assume ~R is true To prove universal statement ( " x) R(x) is true Assume ( \$ x) ~R(x) is true . Try to get a contradiction Conclude that R must be true Try to get a contradiction Conclude that ( " x) R(x) must be true To prove the implication ( " x) P(x) Q(x) is true Assume ( \$ x) P(x) ~Q(x) is true . Try to get a contradiction Conclude that ( " x) P(x) Q(x) must be true
Lecture 9 3 Proof by Contradiction Backward-Forward-table : P1 P Q Q1 Reason Statement Step We only ask forward questions assumption ( \$ x) [P(x) ~Q(x)] contradiction

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Lecture 9 4 Proof by Contradiction Proposition For all positive real numbers x and y, 2 x y y x > + if x y, then Proof by contradiction: Assume ( \$ x \$ y) [P(x,y) ~Q(x,y)] is true: x y and 2 x y y x + P(x,y) Q(x,y) for some positive real numbers x, y
Lecture 9 5 Proof by Contradiction Let x, y be positive real numbers Proof such that x y + xy 2 yx By multiplying xy on both sides, we get +≤ 22 xy2 x y Hence −+ x2 x y y 0 −≤ 2 (x y) 0 and = 2 0 Since square can’t be negative we must have = 0 This happens only if This contradicts the fact the x y Suppose Hence we conclude that + > 2 .

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Lecture 9 6 Proving “There is no” Proposition Proof by contradiction: ~( \$ n) P(n) Symbolic form There is an integer n such that 2n ª 1 mod 4 So we arrive at a contradiction . Assume ( \$ n) P(n) is true There is no integer n such that 2n ª 1 mod 4.
Lecture 9 7 Rational vs Irrational Definition An irrational number is a real number that is not a rational number . A rational number is a real number that can be written as a quotient m/n where m and n are integers, with n > 0. Definition

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Lecture 9 8 Proving Statements involving Irrational Proposition Proof by contradiction: ( " r, s) P(r, s) Q(r, s) Symbolic form i.e. r is rational, s is irrational and r + s is rational for some real numbers r, s.
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lecture09 - MA1100 Lecture 9 Mathematical Proofs Proof by...

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