Let A and B be sets. If A ∩ X = B ∩ X = ϕ and A ∪ X = B ∪ X for some set X, show that A = B.
(Hints A = A ∩ (A ∪ X) , B = B ∩ (B ∪ X) and use Distributive law)
Answers
Solution: While doing such questions first read the given data literally.
1. Here, it's given then common elements in A and X & B and X is nothing (empty or null set).
2. Secondly, the universal set of A and X is same as that of B and X. Now let's solve it numerically.
→ A ∪ X = B ∪ X __(i)
Finding A common out in (i)
→ A ∩ (A ∪ X) = A ∩ (B ∪ X)
(A = A ∩ (A ∪ X); since A is common in A ∪ X)
→ A = (A ∩ B) ∪ (A ∩ X)
→ A = (A ∩ B) ∪ ϕ
→ A = A ∩ B
→ A is Subset of B
On finding B common out in (i)
→ B ∩(A ∪ X) = B ∩(B ∪ X)
→ (A ∩ B) ∪ (B ∩ X) = B
→ B is Subset of A
Since both are subset of each other hence, A = B. Regards :)
Answer:
A ∩ (A ∪ X) = A ∩ (B ∪ X)
(A = A ∩ (A ∪ X); since A is common in A ∪ X)
→ A = (A ∩ B) ∪ (A ∩ X)
→ A = (A ∩ B) ∪ ϕ
→ A = A ∩ B
→ A is Subset of B
On finding B common out in (i)
→ B ∩(A ∪ X) = B ∩(B ∪ X)
→ (A ∩ B) ∪ (B ∩ X) = B
→ B is Subset of A
Step-by-step explanation: