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  1. This is the set \(\{3, 4, 5, \ldots \}\) since we need each element to be a natural number whose square is at least three more than 2. Since \(3^2 - 3 = 6\) but \(2^2 - 3 = 1\) we see that the first such natural number is 3.

  2. We get the same set as we did in the previous part, and the smallest non-negative number for which \(n^2 - 5\) is a natural numbers is 3.

    Note that if we didn’t specify \(n \in \N\) then any integer less than \(-3\) would also be in the set, so there would not be a least element.

  3. This is the set \(\{1, 2, 5, 10, \ldots\}\text{,}\) namely the set of numbers that are the result of squaring and adding 1 to a natural number. (\(0^2 + 1 = 1\text{,}\) \(1^2 + 1 = 2\text{,}\) \(2^2 + 1 = 5\) and so on.) Thus the least element of the set is 1.

  4. Now we are looking for natural numbers that are equal to taking some natural number, squaring it and adding 1. That is, \(\{1, 2, 5, 10, \ldots\}\text{,}\) the same set as the previous part. So again, the least element is 1.

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