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Consider I have two sets that neither one is a subset of the other (for example, the set of prime numbers and the set of odd integers). Is there a specific notation that combines the meanings of both ⊄ and ⊅ ?

Also, is there a notation to mean that two sets are completely distinct and share no elements in common (for example, the sets of odd and even integers)?

≠ seems very inappropriate, and I'd like to avoid it.

Context: My area is actually Computer Science. I'm using this refine my vocabulary in order to teach my students about subnetting (which is basically about sets and subsets of IP addresses), and to write some exercises for them. I'll probably use the words incomparable and disjoint as suggested.

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    $\begingroup$ This can be expressed as $(A \subset B) \downarrow (B \subset A)$, where $\downarrow$ is nor. $\endgroup$
    – user76284
    Oct 27, 2019 at 19:45

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No, there is no special symbol to denote the situation in which neither set is contained in the other. There is a special word, though: we say the two sets are incomparable.

There is also no special symbol to denote the fact that two sets are disjoint; we simply write $A\cap B=\emptyset$.

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Let $\rm\:P\: $ be a poset = partially-ordered set, i.e. a set equipped wth a relation $\rm\le$ that is reflexive, antisymmetric and transitive. Then one says that $\rm\: x\: $ and $\rm\: y\: $ are comparable$\ $ if $\rm\ x \le y\ $ or $\rm\ y \le x\:;\: $ otherwise they are incomparable, sometimes written $\rm\ x||y\:.\:$ A chain$\ $ of $\rm\: P\:$ is a subset in which every two elements are comparable, and dually, an antichain$\ $ is a subset in which every two elements are incomparable.

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    $\begingroup$ (@Juliano: any collection of sets is a poset under the subset relation, which is why this terminology is applicable.) $\endgroup$ Oct 11, 2010 at 21:02
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Not an answer, perhaps an interesting historical curiousity. George Mackey (my professor years ago) hoped to popularize this symbol as a binary relation for disjoint sets:

enter image description here

I think he meant it as a lower case "d" drawn symmetrically.

The screen shot suggests that $\TeX$ doesn't know it.

Edit: This question on $\TeX$ stackexchange provides an update: https://tex.stackexchange.com/questions/554944/looking-for-a-disjoint-symbol

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If subsets $A,B$ are distinct elements of a partition of $U$, then $A \not\sim B$ or $A \not\equiv B$ are perfectly conventional for denoting disjointness as discussed here. (You can make a convenience partition - if $A \cap B = \emptyset$ - as $U = \{A,B, (A \cup B)^c \}$). Similarly, taking any $a \in A, b \in B$ you can write $[a] \not\sim [b]$.

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