Thread: théorème des ensembles

please help me to do this exercice !

http://i23.servimg.com/u/f23/11/67/15/58/88_bmp10.jpg

thanx

Hello, marocker!

I think I've got the second part . . .

Given: .$\displaystyle \begin{array}{ccc}H_1 & = & A \cap B' \cap C'
\\ H_2 & = & (A \cap B) \cup C \end{array}$

Find: .$\displaystyle [1]\;H_1\cup H_2\qquad [2]\;H_1 \cap H_2$

$\displaystyle [2]\;\;H_1 \cap H_2 \;=\;(A \cap B' \cap C') \cap [(A \cap B) \cup C]$

Distribute: .$\displaystyle A \cap B' \cap C' \cap \overbrace{[(A \cup C) \cap (B \cup C)]} $

Rearrange terms: .$\displaystyle \underbrace{[A \cap (A \cup C)]} \cap\, [\underbrace{(B' \cap C')} \cap (B \cup C)] $

DeMorgan's Law: . . . . $\displaystyle A \quad\;\cap\quad\;\underbrace{[(B \cup C)' \cap (B\cup C)]} $

Then: . . . . . . . . . . . . .$\displaystyle \underbrace{A \qquad\quad\cap\qquad\quad \emptyset} $

Therefore:. . . . . . . . . . . . . . . $\displaystyle \emptyset$

and for another question ?