# Example relations: pairwise versus mutual

There are by now several questions on math.se asking about pairwise versus mutual relations, eg:

But confusion remains:

I used to ponder this question some time ago but didn't come up with a satisfactory answer. The problem seems to be that until someone gives a formal treatment of the adjective "pairwise" in mathematics, it must remain a linguistic problem. (user23211)

This phenomenon can be written for a variadic, 2-valued relation $R$ as: $Rab∧Rbc∧Rac∧¬Rabc$.

4x examples:

• Set intersection: "If any two sets intersect it doesn't mean that there exists a common intersection." (Ilya)

• Coprimality: "We have "pairwise coprime" and "coprime". A non-empty set of natural numbers S is pairwise coprime iff $\forall s_1,s_2\in\mathcal S\;\gcd(s_1,s_2)=1$. $S$ is coprime iff $gcd(S)=1$. (user23211)

• Statistical Independence: Pairwise independence of 3 random variables $X, Y, Z$, $P(XY)=P(X)P(Y)$ and $P(XZ)=P(X)P(Z)$ and $P(YZ)=P(Y)P(Z)$ is independent of $P(XYZ)=P(X)P(Y)P(Z)$. The following figure displays this situation graphically with binomial r.v's.

• Topological Linking: The pairwise Gauss linking number of any two links in the 3-component Borromean link is zero. Note, the linking number may be zero even for 2 linked rings, eg Whitehead links. Further, the Gauss linking number is a pairwise relation, as far as I know, it cannot measure the linking of 3 or more components, in fact this motivated development of the Massey product. Nevertheless, topological linking itself is a variadic relation.

Are there other examples from combinatorial geometry and other branches of mathematics?

(Note the Helly theorem in convex geometry states that a sufficient number of intersecting convex sets implies that they all intersect - so that would not be an example of the sought type of relation.)

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What exactly is your question? – Benjamin Dickman Jan 19 '13 at 5:29
@B.D, "big list", what other examples are there of this phenomenon? – alancalvitti Jan 19 '13 at 5:33