Note: Updated based on this.
In my course, my instructor posed the following exercise:
Let $S$ be the subset of $\mathbb R^n$, $S=\{(a_1,a_2,a_3...a_n) | a_2 = \pm a_1, a_3=...=a_n=0 \}$. Define addition and multiplication as follows: For $(a_1,a_2, ..., a_n),(b_1,b_2, ..., b_n) \in S$, define $(a_1,a_2, ..., a_n)+(b_1,b_2, ..., b_n)=(a_1+b_1,a_2+b_2, ..., a_n+b_n)$ and $(a_1,a_2, ..., a_n)(b_1,b_2, ..., b_n)=(a_1 \times b_1,a_2 \times b_2, ..., a_n \times b_n)$, where $+$ and $\times$ are the usual addition and multiplication in $\mathbb R$. Which axioms in the definition of a field are satisfied by $S$? Is $S$ a field and why?
Now, I notice that the given operation $+: S^2 \to \mathbb R^n$ is not closed i.e. its image is not a subset of $S$. In particular it really just says 'define' instead of like giving explicitly the domain and range as $+: S^2 \to \mathbb R^n$. I think this is much like: How do you prove the domain of a function?
I asked my instructor about this and they responded as follows:
(It) is part of the assignment (as to whether or not the operations are closed). If an operation is not closed, which of the rest of the axioms are satisfied?
Question: So basically the trick here is that someone studying this for the 1st time may think of 'define' to really mean $+: S^2 \to S$ instead of $+: S^2 \to \mathbb R^n$ and thus not realise the operation is not closed ?
Cross-posted on maths educator se: https://matheducators.stackexchange.com/questions/24399/can-you-talk-about-the-rest-of-the-field-axioms-when-the-operations-are-not-cl --> I justify the cross-post in that maths education se might respond more to the trick of questions while maths se might respond more to the talking about field axioms when assumptions are violated.
I would argue that "define $+$" does not imply a codomain.
If "closure" is an explicit part of the definition of field (operation) used in the course, then it's not a trick at all, since the definition hands you that as a thing to check.
If "closure" is not an explicit part of the definition, then it's a bit like a trick, but teaches an important lesson in math that not everything you need to check will be listed in a checklist for you. I personally feel that developing the skill of making your own thorough checklists to catch things like this is part of mathematical maturity, and something that the author of the question may hope to help instill.