Disjoint

Stub grade: C

This page is a stub

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:

Proper stub page, created just to minimally document something

Grade: D

This page requires references, it is on a to-do list for being expanded with them.

Please note that this does not mean the content is unreliable, it just means that the author of the page doesn't have a book to hand, or remember the book to find it, which would have been a suitable reference.
The message provided is:

Trust me, disjoint means this

Definition

Let [ilmath]A[/ilmath] and [ilmath]B[/ilmath] be sets. We say "[ilmath]A[/ilmath] and [ilmath]B[/ilmath] are disjoint" if:

• [ilmath]A\cap B=[/ilmath] [ilmath]\emptyset[/ilmath] where [ilmath]A\cap B[/ilmath] denotes the intersection of [ilmath]A[/ilmath] and [ilmath]B[/ilmath]

Disjoint in a set

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

• [ilmath]A\cap B\cap Z=\emptyset[/ilmath]

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

---

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

---

Grade: D

This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.

Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
The message provided is:

Prove these claims

This proof has been marked as an page requiring an easy proof

See also

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