# Thread: De Morgan's laws

1. ## De Morgan's laws

The set (A\B)\C is equal for all sets A,B and C to one of the following sets:

i). A\(B $\cap$ C)
ii). A\(B\C)
iii). A\(B $\cup$ C)

Which one?
This question can be done using truth tables, but it would take a really long time!

I was wondering if there was a way to do it using De Morgan's laws. This is what i've done so far:

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

I'm not sure where to go from here. I tried expanding the other expressions to see if I could get to this, but it hasn't worked.

2. $(A \cap B^c)\cap C^c = A \cap (B^c \cap C^c) = A \cap (B \cup C)^c$

Looks familiar, doesn't it?

3. $
(A \cap B^c)\cap C^c = A \cap (B^c \cap C^c) = A \cap (B \cup C)^c
$

lol, it does (and is):

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

So when you have an expression with all $\cup$'s or $\cap$'s you can change around the brackets? (ie. It's associative?)

4. Originally Posted by Showcase_22
So when you have an expression with all $\cup$'s or $\cap$'s you can change around the brackets? (ie. It's associative?)
Yes, it's associative.