Solution Check: Union/intersect complement

Question

This a true/false question.

For set operations, we always can replace Union by intersection and complement operation.

I think what it is saying is that if A U B, you can swap U with intersect and complement that so they are equal which is obviously false. I just wanted to check if I was understanding what the statement was saying correctly.

You probably want to check that you can replace $A\cup B$ by a combination of $A^c, B^c,$ and $\cap$ — Prahlad Vaidyanathan, Nov 25 '13 at 06:15

score 1 · Answer 1 · answered Jul 06 '18 at 12:56

1

This is true.

Thinking of this like De Morgan's law translated into set theory (using $A^c$ for the complement of $A$):

$$ (A \cup B)^c = A^c \cap B^c $$

And since complement is self-inverse, you get:

$$ A \cup B = (A^c \cap B^c)^c $$

answered Jul 06 '18 at 12:56

Cai

145

score 0 · Answer 2 · answered Nov 25 '13 at 06:17

Here's how I read the question at first glance: For two sets, first take the union, then the complement, so $(A \cup B)^{c}$. This in fact equals $A^{c}\cap B^{c}$. But even if you interpret the question otherwise, that is, if you read it as $A^{c} \cup B^{c}$, it is false.

Solution Check: Union/intersect complement

2 Answers2