Question

Let A and B be sets. \[A \cap X = B \cap X = \varnothing \quad \text{and} \quad A \cup X = B \cup X \quad \text{for some set } X,\ \text{find the relation between } A \text{ and } B.\]
 

• A. \( A = B \)
• B. \( A \cup B = X \)
• C. \( B = X \)
• D. \( A = X \)

Hint:

When two sets have identical unions and disjoint intersections with another set, the two sets must be equal.

Answer 1

The Correct Option is A

We are given the following conditions: 

1. \( A \cap X = B \cap X = \varnothing \) (The intersection of sets \( A \) and \( B \) with \( X \) is the empty set.) 

2. \( A \cup X = B \cup X \) (The union of \( A \) with \( X \) is equal to the union of \( B \) with \( X \).) From the first condition, \( A \cap X = B \cap X = \varnothing \), we can conclude that neither \( A \) nor \( B \) shares any elements with \( X \). 

Therefore, \( A \) and \( B \) must be disjoint with respect to \( X \). 

From the second condition, \( A \cup X = B \cup X \), we can infer that the sets \( A \) and \( B \) must be identical because the union with \( X \) does not change the overall result. If the union of two sets with a third set is the same, the two sets themselves must be equal. 

Therefore, \( A = B \). Thus, the correct answer is \( \boxed{(a) \, A = B} \).