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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?

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  • $\begingroup$ 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. $\endgroup$ – stressed out Feb 14 '19 at 23:56
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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 $A$

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