is the following correct to the question,
the negation should be there are no smallest numbers in every non-epmty subset of the natural numbers, but I am ot sure how to put this in symbol form?
The phrase "There are no smallest numbers in every subset", unfortunately, is ambiguous in English. I hate when people say, "Every group of people does not have a leader" when they mean "Not every group of people has a leader". So, if by "Every subset does not have the smallest element" you mean what this phrase should mean, i.e., that there is not a single subset that has the smallest element, then this is not the negation of "Every subset has the smallest element".
The problem directs you to first write the original statement in quantifiers. Then negating it would be easy.
Well, your first problem is that "there are no smallest numbers in every non-empty subset of the natural numbers" is not correct. The negation of "for all" or "every" is "there exist". The negation of "every non-empty set of natural numbers contains a smallest number" is "there exist a non-empty set of natural numbers that does not contain a smallest number".