How should the word "determine" be interpreted? My question is this, I am asked to prove that two sets A and B determine the same set C iff some condition is satisfied.
How should I understand the above statement? In particular, how should the word determine be interpreted in the above sentence? Here is an example:
Prove that two smooth atlas for a manifold determine the same maximal smooth atlas iff their union is a smooth atlas.
Is there a general interpretation of statements of this type?
Thank you for your time.
Kind regards,
Marius
 A: When you see a phrase in a mathematical discussion of the form "each of these things determines one of those things", that means there's a function lurking around. A function, after all, is something which, given an input from one set, determines an output in another set.
For example, the phrase "every smooth atlas determines a maximal smooth atlas" means that there is a function which inputs a smooth atlas and outputs a maximal smooth atlas. Let's denote this function $\mathcal{X}$; I think of it as "the atlas maximizing function". Suppose one inputs a smooth atlas 
$$\mathcal{A} = \{(U_i,\phi_i) \, \bigm| \, i \in I\}
$$ 
for an $n$-dimensional manifold $M$ (using the notation $U_i \subset M$ and $\phi_i : U_i \to \mathbb{R}^n)$. The output atlas $\mathcal{X}(\mathcal{A})$ is given by throwing in everything that's missing:
$$\mathcal{X}(\mathcal{A}) = \{(V,\psi) \,\bigm|\, \text{for each $i \in I$, the overlap map between $(U_i,\phi_i)$ and $(V,\psi)$ is smooth}\}
$$
Notice that this is indeed a function: the input $\mathcal{A}$ determines the output $\mathcal{X}(\mathcal{A})$.
So in this particular exercise, one's job is to prove that for any smooth atlases $\mathcal{A}$ and $\mathcal{B}$ on the same manifold $M$, we have $\mathcal{X}(\mathcal{A}) = \mathcal{X}(\mathcal{B})$ if and only if $\mathcal{A} \cup \mathcal{B}$ is a smooth atlas for $M$.
