# Names for complement of the union and intersection of two sets.

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?

• You could simplify the former to "The intersection of the complements" and the latter to "The union of the complements" - but there's no need to name everything. – Milo Brandt Mar 8 '15 at 22:26
• @Meelo - Aside from the fact that you (like i before i fixed the Q) have them reversed, you are not actually simplifying the description, merely providing another permutation. Appreciated nonetheless. – Martin F Mar 10 '15 at 15:11

## 2 Answers

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.

• I wondered why you were initially "merely" restating my question -- until i realized my serious typo, NAND - NOR reversal, now fixed, in the question ;-) – Martin F Mar 9 '15 at 2:23