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"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. (?)

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    Many people use "disjoint" and "pairwise disjoint" synonymously. I have seen "free" be used for families with nonempty intersection (as in "free ultrafilter"). The case of independence and pairwise independence is essentially about local and global properties.2012-04-16
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    I am not entirely sure why this is tagged foundations. To me that tag suggests http://en.wikipedia.org/wiki/Foundations_of_mathematics This feels to me more a question about (notation) or (terminology).2012-04-17
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    @WillieWong : Maybe it's because as soon as I suspect category theory of being involved, I think "foundations". Maybe that's obsolete by 75 years or so?2012-04-17
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    I agree with Willie that the [foundation] tag seems off, and personally I am not sure there is a real need for this tag too.2012-04-17
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    ....I guess another reason for the "foundations" tag is that this seemed like something basic that could apply to virtually any area of mathematics.2012-04-18
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    But that is precisely my point: "foundations" means something quite specific about how mathematics is built. You seem to be using that word in a different way from how I would.2012-04-24
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    @Willie : Maybe at this point you should be specific about what you take "foundations" to mean. Do you mean something like encoding everything within ZFC?2012-04-24
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    @MichaelGreinecker : Just in case some people haven't noticed, your point about terminology isn't strictly relevant to my question, unless maybe you just meant I should phrase it differently.2012-04-24
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    That's precisely why I included a Wikipedia link in my first comment. So yes, I mean something like encoding everything within ZFC, or some other formal system.2012-04-24
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    "Pairwise coprime" is another case where "pairwise" strengthens.2012-04-26
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    ....and now I've added "pairwise coprime" to the question.2012-04-26
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    I don't think I've ever seen "disjoint" used (for sets) in any other way than as a synonym for "pairwise disjoint".2012-04-27
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    Porbably within a couple of days I'll decide whether to "accept" one of the answers below . . . . .2012-04-30

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