Rational odds
Is there an assignment of the rational numbers to even and odd such that:
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The sum of two odd numbers is even
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The sum of two even numbers is even
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The sum of an even and odd number is odd
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Integers maintain their usual assignment as even or odd (for instance, 3 must be odd and -4 must be even).
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1 solution
Consider the assignment of 1/2. Since 1/2+1/2=1 and 1 must be odd, there is no possible assignment of 1/2.