# What does pairwise disjoint mean?

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?

• Pairwise disjoint means that any pair of the sets has empty intersection, i.e no overlap in elements. – rubikscube09 Mar 9 '19 at 18:32
• 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, $b=g$). – Chris Leary Mar 9 '19 at 18:34
• – Shivering Soldier Mar 9 '19 at 18:45

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.