Question

Identify the correct statement

• A. \( A \cup A' = \emptyset \)
• B. \( A - B = A' \cap B \)
• C. \( (A \cup B)' = A' \cup B' \)
• D. \( A \subseteq B \Rightarrow B' \subseteq A' \)

Hint:

Remember De Morgan's laws and complement properties carefully; complement reverses subset relations.

Answer 1

The Correct Option is D

Step 1: Check option (A).
\[ A \cup A' = U \neq \emptyset. \]
So, option (A) is incorrect.

Step 2: Check option (B).
\[ A - B = A \cap B'. \]
But option says \(A' \cap B\), which is incorrect.

Step 3: Check option (C).
Using De Morgan's law:
\[ (A \cup B)' = A' \cap B'. \]
But option says union instead of intersection, so it is incorrect.

Step 4: Check option (D).
If:
\[ A \subseteq B, \]
then every element of \(A\) is in \(B\).
Taking complements reverses inclusion:
\[ B' \subseteq A'. \]

Step 5: Verify logically.
If an element is not in \(B\), then it cannot be in \(A\), hence it belongs to \(A'\).

Step 6: Conclusion from options.
Only option (D) satisfies correct set identity.

Step 7: Final conclusion.
\[ \boxed{A \subseteq B \Rightarrow B' \subseteq A'} \]