TITLE: When does "pairwise" strengthen and when does it weaken?
QUESTION [12 upvotes]: "Pairwise disjoint" is stronger than "disjoint"; it sometimes happens that $\displaystyle\bigcap\limits_{i\in I} A_i=\varnothing$ but for every $i,j$, or at least for some, one has $A_i \cap A_j\ne\varnothing$.
Likewise, "pairwise coprime" is stronger than "coprime".
"Pairwise independent" (in the probabilistic sense) is weaker than "independent".  For example, suppose $X_1,\ldots,X_n$ are independent and uniformly distributed on the sphere $S^n$; then the great-circle distances $d(X_i,X_j)$ and $d(X_k,X_\ell)$ are independent if $\{i,j\}\ne\{k,\ell\}$ even if $i=k$, but $d(X_i,X_j), d(X_j,X_k), d(X_k,X_i)$ are not independent even though each of the three pairs of these three random variables is a pair of independent random variables.
Question: Is there a somewhat general rule that says for which sort of X the qualified "pairwise X" is stronger than "X" and for which sort it is weaker than "X"?
Here's a guess: Some sort of category-theoretic viewpoint can make some kind of sense of this. (?)

REPLY [7 votes]: It depends on which way the implication goes between the pairwise property and the property of the whole set. 
More generally, sometimes if a property $P(S)$ holds for an entire set $S$, then we are assured that it also holds for all subsets of $T$ of $S$, such as pairs. In other words:
$$P(S) \implies \forall T:T \subset S \to P(T)$$
The implication sometimes goes the other way: if we know the property holds for subsets of $S$, then it holds for $S$:
$$\forall T:T \subset S \to P(T) \implies P(S)$$
Or it could be the case that the property of the whole is independent of the property of subsets.
It could also be the case that the relationship is actually $\Leftrightarrow$. For instance an equivalence relation is like this. If elements of a set are all equal pairwise, then necessarily, all elements are equal to each other, and vice versa.
Whether the subset-based proposition is weaker than the whole-set one depends on whether they are connected together logically, and, if so, how: which way does the implication point, or are they logically equivalent.
(Note that this is a very general idea; for instance I'm glossing over situations when, say, it is understood that the pairs in "pairwise" are formed by the cartesian product of the set with itself, and are tuples, rather than sets. The same principles apply, if not the exact formulas.)