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What is a good name for $ (A \cup B)^c $ or the complement of the union of two sets? Not union? NOR? Union complement?
And what is a good name for $ (A \cap B)^c $ or the complement of the intersection of two sets? Not intersection? NAND? Intersection complement?
As $ (A \cup B)^c = A^{c} \cap B^c $ and $ (A \cap B)^c = A^{c} \cup B^c $, I'd say "complements intersection" and "complements union", respectively. There's no proper term though.
NOR is usually $(A\cup B)^c$. NAND usually means $(A\cap B)^c$.
NAND and NOR are more commonly used for binary operations, particularly talking about logic gates, but they extend to set definitions.
The expression $A\cup B$ is the union, and thus is "OR" - the elements of $A\cup B$ are either elements of $A$ OR elements of $B$. The elements of $(A\cup B)^c$ are thus neither elements of $A$ NOR elemenents of $B$.
NAND and NOR are interesting in that everything can be defined in terms of them.
For example, if you have $A\star B$ defined as NOR, then $$\begin{align}A^c=&A\star A \\A\cap B&=A^c\star B^c = (A\star A)\star(B\star B)\\ A\cup B&=(A^c \cap B^c)^c =\text{ something horrific} \end{align}$$
The fundamental disadvantage of NAND and NOR is the lack of associativity.
