Disjoint

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Definition

Let $A$ and $B$ be sets. We say "$A$ and $B$ are disjoint" if:

• $A\cap B=$ $\emptyset$ where $A\cap B$ denotes the intersection of $A$ and $B$

Disjoint in a set

Let $Z$ be a set and let $A$ and $B$ be sets (with no other requirements), then we say "$A$ and $B$ are disjoint in $Z$" if:

• $A\cap B\cap Z=\emptyset$

Comments on "disjoint in a set"

There are 2 ways to think about it that show intuitively what we mean by "disjoint in a set":

1. $(A\cap Z)\cap(B\cap Z)=\emptyset$ is probably the most natural, we're saying that the parts of $A$ and $B$ actually in $Z$ must be disjoint
   • This is easily seen to be equivalent to the above definition
2. $A\cap B\subseteq Z^\complement$ - where $Z^\complement$ denotes the set complement of $Z$, (which may not always be defined/make sense) and why we have the other form

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TODO: I've proved $(A\cap B\cap Z=\emptyset)\iff(A\cap B\subseteq Z^\complement)$ on paper, It seems that we infact have: $(A\cap B=\emptyset)\iff(A\subseteq B^\complement)$, maybe they should get pages....

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See also

• Pairwise disjoint
• Non-empty
  • Non-empty in a set
• Disjoint in a set (links to above section for "disjoint in a set")