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MATH135 - Assignment3 Solutions

# MATH135 - Assignment3 Solutions - MATH 135 Fall 2007...

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Unformatted text preview: MATH 135 Fall 2007 Assignment #3 Due: Wednesday 03 October 2007, 8:20 a.m. Hand-In Problems 1. Throughout this problem, the universe of discourse is R , the real numbers. (a) Is “ ∀ x ∀ y, x 2 + 1 = y 2 ” TRUE or FALSE? Explain. (b) Is “ ∃ x ∃ y, x 2 + 1 = y 2 ” TRUE or FALSE? Explain. (c) Is “ ∃ x ∀ y, x 2 + 1 = y 2 ” TRUE or FALSE? Explain. (d) Is “ ∀ x ∃ y, x 2 + 1 = y 2 ” TRUE or FALSE? Explain. (e) Is “ ∀ y ∃ x, x 2 + 1 = y 2 ” TRUE or FALSE? Explain. 2. Express the following English language statements using quantifiers. Throughout this problem, the universe of discourse is Z , the integers. (a) “Every integer is 7 more than some integer.” (b) “There is an integer that divides into every integer.” 3. For each pair a and b , state the quotient and remainder when a is divided by b . (a) a = 387, b = 11 (b) a =- 387, b = 11 (c) a = 387, b =- 11 (d) a =- 387, b =- 11 4. (a) Determine gcd(987 , 320). (b) Determine integers x and y such that 987 x + 320 y = gcd(987 , 320). 5. (a) Determine gcd(2193 , 1008). (b) Determine integers x and y such that 2193 x + 1008 y = gcd(2193 , 1008). 6. Suppose that a and b are positive integers and that when a is divided by b , a quotient of q and remainder of r are obtained, with r > 0. Determine the possible quotients and remainders when 3 a is divided by b and the conditions on r that give each possibility, making sure to justify that your answer is correct using the statement of the Division Algorithm. 7. Suppose a, b 1 , b 2 , . . . , b n ∈ Z with a | b 1 , a | b 2 , . . . , a | b n . Prove by induction on n that a | b 1 x 1 + b 2 x 2 + · · · + b n x n for all integers x 1 , x 2 , . . . , x n . 8. As with the integers, there is a Division Algorithm for polynomials: Suppose that f ( x ) and g ( x ) are polynomials in x with coefficients from Q (or R ) and g ( x ) is not the zero polynomial. Then there exist unique polynomials q ( x ) and r ( x ) such that f ( x ) = q ( x ) g ( x ) + r ( x ) where r ( x ) = 0 or the degree of r ( x ) is less than the degree of g ( x ). Note that the condition on r ( x ) is quite similar to the analogous condition on r in the integer Division Algorithm. For example, if f ( x ) = x 4 + x 3 + 2 x- 1 and g ( x ) = x 2- x- 2, we can perform long division: x 2 + 2 x + 4 x 2- x- 2 | x 4 + x 3 + 0 x 2 + 2 x- 1 x 4- x 3- 2 x 2 2 x 3 + 2 x 2 + 2 x 2 x 3- 2 x 2- 4 x 4 x 2 + 6 x- 1 4 x 2- 4 x- 8 10 x + 7 so the quotient is q ( x ) = x 2 + 2 x + 4 and the remainder is r ( x ) = 10 x + 7....
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MATH135 - Assignment3 Solutions - MATH 135 Fall 2007...

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