1. ## Set proof

Hello guys, I'm having some trouble proving this (it's from the introduction of Apostol's calculus book). Thx

(a) Prove that one of the following two formulas is always right and the other one is sometimes wrong:

(i) $\text A - (B - C) = (A - B) \cup C$
(ii) $\text A - (B \cup C) = (A - B) - C$

(b) State an additional necessary and sufficient condition for the formula which is sometimes incorrect to be always right.

2. Originally Posted by yannick
Hello guys, I'm having some trouble proving this (it's from the introduction of Apostol's calculus book). Thx

(a) Prove that one of the following two formulas is always right and the other one is sometimes wrong:

(i) $\text A - (B - C) = (A - B) \cup C$
(ii) $\text A - (B \cup C) = (A - B) - C$

(b) State an additional necessary and sufficient condition for the formula which is sometimes incorrect to be always right.
i'll do the second one. i leave the first to you (it is also the one that is sometimes wrong). follow similar steps to what i do here and apply it to the first one. see if you can spot what the problem is that causes it to be sometimes wrong

(ii) $A - (B \cup C) = A \cap (B \cup C)'$ ...................by definition

........................ $= A \cap (B' \cap C')$ ..................by DeMorgan's laws

........................ $= A \cap B' \cap C'$

........................ $= (A \cap B') \cap C'$

........................ $= (A - B) \cap C'$

........................ $= (A - B) - C$

3. Thx for the help, but in my book there isn't that definition and the de Morgan's law. Any other suggestion?

thx again

4. Originally Posted by yannick
but in my book there isn't that definition and the de Morgan's law. Any other suggestion?
Well, tell what your textbook does have in it to use.

5. I'm usnig Calculus vol 1 by Apostol

6. I forget to say that the definition he used is not a problema to understand, even if this is not mentioned on the book, but the de Morgan's law is not present

7. Originally Posted by yannick
I forget to say that the definition he used is not a problema to understand, even if this is not mentioned on the book, but the de Morgan's law is not present
as Plato said, tell us, what can you do. what tools do you have at your disposal to taclke these problems. in any case, DeMorgan's laws are laws you should know, whether your professor covered them or not

8. Originally Posted by Jhevon
as Plato said, tell us, what can you do. what tools do you have at your disposal to taclke these problems. in any case, DeMorgan's laws are laws you should know, whether your professor covered them or not
Well I'm studying by my self so I will try to study this laws. Actually I've just bought a book called "How to prove it" and in the introduction to logic part there are this de morgan's laws so I will try to prove it after reeding this part, maybe they are pre-requisites for the book.

In the set theory chapter of Apostol's calculus book there are just the basic laws, union, intersection, diference of sets, associative, commutative and distributive propriety.