So the question is write down a definition for it means to be closed under an operation. I said that a set is closed under an operation if that operation returns an element in the set when evaluated with other elements in the set.
The sets that we have to test are
1) the set of numbers or the form $M+N\sqrt{3}$ where $M,N$ are any integers under multiplication. I don't really know what set this is defining.
2) The set of irrationals under multiplication. This is false because $\sqrt{2}\times\sqrt{2}=2$, which is a rational number.
3)Let $\mathbf{F}$ be an ordered field and let $P$ be the set of positive elements. Let $N$ be the set of non-positive, non-zero elements in $\mathbf{F}$ . In other words $N=\mathbf{F}\backslash (P\cup\{0\})$. I first have to prove that it is closed under addition and then determine whether N is closed under multiplication. For this one I think it is not closed under addition, but it is closed under multiplication, but I don't know how to formally prove it
The first part, the set is $\{ M + N\sqrt 3 |M,N \in \mathbb Z\}$. Think of it as a formal sum, like the complex numbers, if you have seen complex numbers.
This is closed under both addition and multiplication, since $(M + N \sqrt 3 ) + (M' + N'\sqrt 3) = (M+M') + (N + N')\sqrt3$, and both $M+M'$ and $N+N'$ are integers, so we have an element of the set. Similarly $(M+N\sqrt 3) (M'+N'\sqrt 3) = (MM' + 3NN') +(MN' + M'N)\sqrt3$, and again the coefficients are integers.
With regards to three, you can use $\mathbb R$ as an example. What happens when you multiply two negative elements, and what happens when you sum two negative elements?