Question

If A and B are any two non-empty sets, then \[ [(A \cap (A \cup B)^C)] \cup [(A \cap B)\cup(A\cap B^C)] = \]

• A. \(\emptyset\)
• B. \(B\)
• C. \(A\)
• D. \(A\cap B\)

Hint:

Remember the distributive identity: \[ (A\cap B)\cup(A\cap C)=A\cap(B\cup C). \] It is frequently used in simplifying set expressions.

Answer 1

The Correct Option is C

Step 1: Simplify the first term. Using De Morgan's law, \[ (A\cup B)^C=A^C\cap B^C \] Therefore, \[ A\cap(A\cup B)^C = A\cap A^C\cap B^C \] Since \[ A\cap A^C=\emptyset \] we get \[ A\cap(A\cup B)^C=\emptyset \]

Step 2: Simplify the second term. \[ (A\cap B)\cup(A\cap B^C) \] Taking \(A\) common, \[ = A\cap(B\cup B^C) \] Since \[ B\cup B^C=U \] \[ =A\cap U \] \[ =A \]

Step 3: Combine both results. \[ \emptyset\cup A = A \] Hence, \[ \boxed{A} \]