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Homework 4 Solution

# Homework 4 Solution - MATH 74 HOMEWORK 4(DUE WEDNESDAY...

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MATH 74 HOMEWORK 4 (DUE WEDNESDAY SEPTEMBER 26) 1. (c). [Interesting note: as of this writing, I do not think it is known whether the statement is true or false.] 2. (d). 3. Suppose the statement is false. Then there is a rational number x and an irrational number y with the property that the number z = x + y is rational. But then y = z - x , and a difference of rational numbers is rational. This contradicts the fact that y is irrational, proving the statement. 4. Suppose the statement is false. Then there is a nonzero rational number x and an irrational number y with the property that the number z = xy is rational. But then y = z x (which makes sense as x is nonzero) is a quotient of rational numbers. We know that a quotient of rational numbers is rational, so y is rational. This contradicts the fact that y is irrational, proving the statement. 5. If the assertion is false, then there are unequal positive integers x and y satisfying x 2 + xy = 2 y 2 . But then x 2 + xy - 2 y 2 = 0, and factoring we conclude ( x - y )( x + 2 y ) = 0 .

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Homework 4 Solution - MATH 74 HOMEWORK 4(DUE WEDNESDAY...

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