Section1.7

# Section1.7 - b and c are real numbers and a 6 = 0 then...

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Section 1.7 Proof Methods and Strategy: 1. Prove that n 3 3 n if n is a positive integer less than 5. 2. Question 28 on Page 103: Prove that there is no solution in integers x and y to the equation 2 x 2 + 5 y 2 = 14.

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3. Definition: The absolute value of x is defined as | x | = ( x if x 0, - x if x < 0. Question 5 on Page 102: Prove the triangle inequality which states that if x and y are real numbers then | x | + | y | ≥ | x + y | .
Existence Proofs: xP ( x ) 1. Prove that there exist rational numbers x and y sucht that x y is irra- tional. 2. Question 11 on Page 103: Prove that there is a rational number x and an irrational number y such that x y is irrational.

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3. A new quantifier: ! xP ( x ) is ”There is exactly one (a unique) x such that P ( x ) . ! xP ( x ) ≡ ∃ x { P ( x ) [ y ( y 6 = x → ¬ P ( y ))] } Question 14 on Page 103: Show that if
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Unformatted text preview: b and c are real numbers and a 6 = 0, then there is a unique solution of the equation ax + b = c . 4. Another quantifcation: Question 30 on Page 103: Prove that there are infnitely many solutions in positive integers x , y and z to the equation x 2 + y 2 = z 2 . 5. Question 20 on Page 103: Show that if x is a nonzero real number then x 2 + 1 x 2 ≥ 2. 6. Question 33 on Page 108: Prove that if x is irrational and x ≥ 0 then √ x is irrational. 7. Question 10 on Page 102: Show that the product of two of the numbers 65 1000-8 2001 +3 177 , 79 1212-9 2399 +2 2001 , and 24 4793-5 8192 +7 1777 is nonnegative....
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