Can you define what it means that a set of functions has a common property?

Suppose $$\mathcal{A}$$ is a set of functions with a common property. Does this mean that the property is shared pairwise or that the property is shared amongst all elements of $$\mathcal{A}$$?

For example, what does it mean that a set of polynomials do not have a common zero?

• This is a very good question. I think it's impossible to say which one is intended by the author. I would assume by default that when it's said a set of polynomials do not have a common zero, it means that there is no $a$ such that $p(a)=0$ for all $p$ because in mathematics we usually deal with quantifiers $\forall$ and $\exists$. The pairwise interpretation is also equally likely. So, it's really ambiguous. – stressed out Feb 14 '19 at 23:56

1 Answer

Generally, this means that all elements of a set share that property, but it depends on the context a bit. For the polynomial zero question, it is assumed that this means that there is no value $$x$$ for which $$f(x) = 0$$ for all $$f$$ in set $$A$$