Solve Set Proof Problems Homework Statement

[Ayushi160695]

Homework Statement

Can anyone please help me solve these questions?
(1) Prove that (A-B) - (B-C) = A-B
(2)Simplify (A-( A N B)) N (B-(ANB))
(3) Simplify ( ( A N ( B U C)) N ( A-B)) N ( B U C')
(4)Use element property and algebraic argument to derive the property
(A-B) U (B-C) = (A U B) - (B N C)
(5) Derive the set identity A U (A NB) = A
(6) Derive the set identity A N ( A U B) = A
N stands for intersection.
Thank you for your time.

Homework Equations

A-B = A N B'- Set Difference rules
A N ( A U B) - Distributive rule : (A N A) U (A N B)
(A' N B') = (A U B)' - De Morgans Law

The Attempt at a Solution

Distributive[/B] rules A N (A U B) = ( A N A) U ( A N B) = A U (A NB) then I don't know how to go there, if i continue with associative rule I simply revert back to the initial step? Same for question 5. For Q 1 (A N B') N ( B' N C) and then I don't know how to go from there.
Q 2 and 3 got me confused with all the brackets to be honest...I don't even know which rules to use? Any hint?

 

[PeroK]

Have you ever seen the following definition that two sets are equal:

Sets $A$ and $B$ are equal if $x \in A \iff \ x \in B$

 

[Ayushi160695]

Have you ever seen the following definition that two sets are equal:

Sets $A$ and $B$ are equal if $x \in A \iff \ x \in B$

I actually managed to get the answer to Q 5 and 6
Q 5 First we state that for all subset of B of the universal set, U U B = U and then we intersect both sides by A.
Q 6 We state that for all subset if B of the universal set, Null set = Null Set intersect B and then take union of both sides with A.
It was quite easy, I was quite dumb for not noticing before and now for the other questions...