1
$\begingroup$

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?

$\endgroup$
2
  • 2
    $\begingroup$ 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. $\endgroup$ – Milo Brandt Mar 8 '15 at 22:26
  • $\begingroup$ @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. $\endgroup$ – Martin F Mar 10 '15 at 15:11
1
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ I wondered why you were initially "merely" restating my question -- until i realized my serious typo, NAND - NOR reversal, now fixed, in the question ;-) $\endgroup$ – Martin F Mar 9 '15 at 2:23

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.