# Thread: Algebraic laws to simplify

1. ## Algebraic laws to simplify

For any sets A and B, use algebraic laws of sets to simplify
(a) [ A - (A-B)] - B
(b) A - [ B - (A-B)]

2. ## Re: Algebraic laws to simplify

Use $X \setminus Y = X \cap Y^c$ to convert the set minus into intersection/union of sets

3. ## Re: Algebraic laws to simplify

Originally Posted by Bingk
Use $X \setminus Y = X \cap Y^c$ to convert the set minus into intersection/union of sets
How to put the set minus into the set?

4. ## Re: Algebraic laws to simplify

I don't understand your question ...

Example:
$X \setminus ( Y\setminus Z) = X \ ( Y \cap Z^c) = X \cap (Y \cap Z^c)^c = X \cap (Y^c \cup Z) = \\ (X \cap Y^c) \cup (X \cap Z)$

The last line is from distribution (you can check Algebra of sets - Wikipedia, the free encyclopedia for more "shortcuts")