Question

Prove for any set X and for any subsets A and B of X, the set A can be written as a disjoint union in the form A = (A ∩ B) ∪ (A ∩ B^c).

a) A ∩ B^c = \emptyset\
b) A ∩ B = A
c) A ∩ B = \emptyset\
d) A ∩ B^c = A

Answer 1

For any set X and subsets A and B of X, the set A can be written as a disjoint union in the form (A ∩ B) ∪ (A ∩ Bc).

Step-by-step explanation:

We can prove that for any set X and for any subsets A and B of X, the set A can be written as a disjoint union of (A ∩ B) ∪ (A ∩ Bc). Here's a step-by-step explanation:

1. Let's start by understanding what each term means.
   • A ∩ B represents the set of elements that are common to both A and B.
   • A ∩ Bc represents the set of elements that are in A but not in B.
2. To prove the given statement, we need to show that (A ∩ B) and (A ∩ Bc) are disjoint, meaning they have no elements in common.
3. If x ∈ (A ∩ B) ∪ (A ∩ Bc), then x is either in (A ∩ B) or in (A ∩ Bc).
   • If x ∈ (A ∩ B), then x is in both A and B.
   • If x ∈ (A ∩ Bc), then x is in A but not in B.
4. Since x cannot be in both (A ∩ B) and (A ∩ Bc) at the same time, these two sets are disjoint.
5. Therefore, A = (A ∩ B) ∪ (A ∩ Bc) can be written as a disjoint union.