Which of the following Boolean algebra rules is correct ?

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NIELIT Scientific Assistant IT 5 Dec 2021 Official Paper
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  1. A.A̅ = 1
  2. A + AB = A + B
  3. A(A + B) = B
  4. A + A̅B = A + B

Answer (Detailed Solution Below)

Option 4 : A + A̅B = A + B
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Detailed Solution

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The correct answer is option 4: A + A̅B = A + B

Key Points

  • A + A̅B = A + B is a valid Boolean identity derived using the Distributive Law and the Law of Complementation.
  • It is a known simplification rule in Boolean algebra.
  • Proof:
    • A + A̅B = (A + A̅)(A + B) (Distributive Law)
    • (A + A̅) = 1 (Complement Law)
    • ⇒ 1.(A + B) = A + B

Additional Information

  • Option 1 – A.A̅ = 1: ❌ Incorrect. A.A̅ = 0 (by Complement Law)
  • Option 2 – A + AB = A + B: ❌ Incorrect. A + AB = A (by Absorption Law)
  • Option 3 – A(A + B) = B: ❌ Incorrect. A(A + B) = A (by Absorption Law)

Hence, the correct answer is: option 4: A + A̅B = A + B

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