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Say I have a set, $\{a,mc,d\}$. Would $\{a,mc\}$, $\{a,d\}$, and $\{mc,d\}$ be pairwise disjoint? And what about $\{b,g\}$, is $\{b,b\}$, $\{g,g\}$ pairwise disjoint?

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    $\begingroup$ Pairwise disjoint means that any pair of the sets has empty intersection, i.e no overlap in elements. $\endgroup$ Mar 9, 2019 at 18:32
  • $\begingroup$ Typically, pairwise disjoint means that no two sets in the collection have a member in common. The answer to your first question is no, in fact each pair of distinct subsets has a common element. For your second question, you may as well ask if $\{b\}$ and $\{g\}$ are disjoint, which they are (unless, of course, $b=g$). $\endgroup$ Mar 9, 2019 at 18:34
  • $\begingroup$ Related -math.stackexchange.com/q/3140228/588038 $\endgroup$
    – cqfd
    Mar 9, 2019 at 18:45

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Say we have some collection of sets $S_1, S_2, S_3, ..., S_n$. We say that these sets are $k$-wise disjoint if any $k$ of these sets have empty intersection, that is $S_{x_1} \cap S_{x_2} \cap \dots \cap S_{x_k} = \varnothing$ for any choice of $x$'s. Note that the smaller $k$ is, the stronger the statement.

Pairwise disjoint is another way of saying $2$-wise disjoint, that is if you pick any two of these sets then they have empty intersection.

A related term is mutually disjoint, which also means $n$-wise disjoint. This just means that there does not exist a single element found in all of these sets.

Example: Consider the sets $\{1, 2\}, \{2, 3\}, \{3, 4\},$ and $\{4, 1\}$. They are not pairwise disjoint. For example, $\{1, 2\}$ and $\{2, 3\}$ share an element. However, they are $3$-wise disjoint (check yourself) and therefore also $4$-wise disjoint.

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  • $\begingroup$ If i understand correctly, it follows that a collection is pairwise disjoint just in case it is $n$-wise disjoint. Is this true? $\endgroup$
    – Marmitrob
    Dec 3, 2023 at 19:47

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