Final answer:
An example of a one-to-one but not onto function is f(x) = 2x, and an example of an onto but not one-to-one function is g(x) = ⌊x/2⌋. An example of a function that is both onto and one-to-one is h(x) = x + 1. An example of a function that is neither one-to-one nor onto is k(x) = 0.
Step-by-step explanation:
(a) One-to-one but not onto:
An example of a function from N (the set of natural numbers) to N that is one-to-one but not onto is the function f(x) = 2x. This function pairs each natural number with its double. For example, f(1) = 2, f(2) = 4, f(3) = 6, and so on. It is one-to-one because each input has a unique output, but it is not onto because there are natural numbers in the codomain (range) that do not have a preimage in the domain.
(b) Onto but not one-to-one:
An example of a function from N to N that is onto but not one-to-one is the function g(x) = ⌊x/2⌋, where ⌊x⌋ represents the greatest integer less than or equal to x. This function maps each natural number to its floor division by 2. It is onto because every natural number has a preimage, but it is not one-to-one because multiple inputs can map to the same output. For example, g(2) = g(3) = 1.
(c) Both onto and one-to-one:
An example of a function from N to N that is both onto and one-to-one (but different from the identity function) is the function h(x) = x + 1. This function maps each natural number to its successor. It is onto because every natural number has a preimage, and it is one-to-one because each input has a unique output.
(d) Neither one-to-one nor onto:
An example of a function from N to N that is neither one-to-one nor onto is the function k(x) = 0. This function maps all natural numbers to 0. It is not one-to-one because all inputs map to the same output, and it is not onto because there are natural numbers in the codomain that do not have a preimage.