# Is there generally understood set theory term for $A \cap B \setminus C$

I'm interested in formalising/documenting a program plotting Venn charts (well UpSet plots actually). Is there a term that can unambiguously refer to the coloured partition of the visualisation below (obtained from Wolfram alpha from $$A \cap B \setminus C$$) in relation to $$A$$ and $$B$$? It was previously referred to as distinct intersection of $$A$$ and $$B$$ (because it does not include the intersection of $$A$$ and $$B$$ with any other set) - is there a formal term for this concept? Edit: in general I am interested in a case of $$n$$ sets. I am looking for a generic name for a region of an intersection of $$k$$ sets ($$k \leq n$$) such that it is exclusive to the sets forming the intersection this is it excludes overlap with any other set that do not for this $$k$$ intersection.

• $(A\cap B)\setminus C = A\cap (B\setminus C)$, so the expression is technically unambiguous. It is still bad form in my opinion to write this without parentheses since it is not immediately apparent to a beginner that this should be the case. Dec 18 '20 at 17:59
• As for if there is a specific name for that set or operation... none that I am aware of beyond the obvious "intersection of $A,B$ and $C^c$" or "Intersection of $A$ and $B$ without $C$" etc... I hesitate to say something like "The region simultaneously occupied by $A$ and $B$ and no others" in general because the image only alludes to the additional set $C$ but does not preclude there existing other unpictured sets which may also occupy a portion of that region. Dec 18 '20 at 18:05

No, there is not any commonly used term for this. Simply writing it as $$(A\cap B)\setminus C$$ or $$A\cap B\cap C^c$$ (with $${C}^c$$ referring to the complement of $$C$$) would be a reasonable choice. If you want to refer to it in words, you could say something like "the region that is in $$A$$ and $$B$$ but not $$C$$", or "the intersection of $$A$$ and $$B$$ and the complement of $$C$$". You could also say something like "the region that is only in $$A$$ and $$B$$". I would strongly recommend against "distinct intersection"; that's not the way "distinct" is normally used in mathematics.
• Thank you. I like distinct because it is a single word that can be used as an argument value in a programming implementation that I am preparing here to achieve a result demonstrated there. Would you recommend a different choice of up to three words for a general case? Should I edit my question to reflect that I am interested in a general case of n sets, and the A, B, C are just examples? Dec 19 '20 at 0:50
• I'm thinking about this region as being a partition of $A \cap B$ that is exclusive to $A$ and $B$ but I am afraid of using exclusive either as it has yet another meaning in set theory. Dec 19 '20 at 0:56