Proof of the property using set theory:
1. Definition of Set Minus:
- The set A minus B (A - B) is defined as the set of all elements in A that are not in B.
2. Expanding A minus (B minus C):
- A minus (B minus C) means we first take the set difference of B and C, and then take the set difference of A and the result.
- This can be rewritten as A intersection (B complement C) using set theory.
3. Expanding (A minus B) union (A intersection C):
- (A minus B) union (A intersection C) means we first take the set difference of A and B, and then take the intersection of A and C, and finally take the union of the two results.
- This can be rewritten as A intersection (B complement) union A intersection C using set theory.
4. Proving Equality:
- To prove the equality, we need to show that A intersection (B complement C) is equal to A intersection (B complement) union A intersection C.
- By using the distributive property of set intersection over union, we can show that the two expressions are equal.
5. Conclusion:
- Therefore, by using the property of sets and set theory, we have proved that A minus (B minus C) is equal to (A minus B) union (A intersection C).