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Consider the mapping f from the natural numbers to the integers given by f(n) = (n - 1)/2 for odd n, and f(n) = -n/2 for even n. Which of the following best describes f?

1. neither injective nor surjective
2. injective but not surjective
3. surjective but not injective
4. both injective and surjective

Correct answer: neither injective nor surjective

Solution

The function f is not injective because multiple natural numbers can map to the same integer (e.g., f(1) = 0 and f(2) = -1), and it is not surjective because not all integers are covered (e.g., there is no natural number n such that f(n) = -1.5).

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