Problem 14 · 2013 Math Kangaroo
Medium
Logic & Word Problems
casework
Consider the following statement about a function \(f : \mathbb{Z} \to \mathbb{Z}\) defined for all integers x: “For every even x, \(f(x)\) is even.” What is the negation of this statement?
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Answer: D — There is a number x for which \(f(x)\) is odd.
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Hint 1 of 2
Negating 'for every …' turns it into 'there exists … that fails'.
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Hint 2 of 2
The failure is: an x where f(x) is odd.
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Approach: negate a universal statement
- The claim is 'for every even x, f(x) is even.'
- Its negation asserts the existence of some x for which f(x) is odd.
- The matching choice is D.
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