Boolean Algebra Questions

Question 1: Simplify the Boolean expression A + A'B.

Solution: A + B

• A + A'B is simplified using the Distributive Law: A + A'B = A + B.
• This happens because A + A' = 1, and 1B = B.

Question 2: Simplify the Boolean expression A(A + B).

Solution: A

• Using the Distributive Law: A(A + B) = AA + AB.
• Since AA = A, the expression becomes A + AB.
• Using Absorption Law: A + AB = A.

Question 3: Simplify the Boolean expression A' + AB.

Solution: A' + B

• Using Distributive Law: A' + AB = (A' + A)(A' + B).
• Since A' + A = 1, the expression simplifies to 1(A' + B) = A' + B.

Question 4: Simplify the Boolean expression AB + A'B.

Solution: B

• Using the Distributive Law: AB + A'B = B(A + A').
• Since A + A' = 1, the expression simplifies to B.

Question 5: Simplify the Boolean expression (A + B)(A + B').

Solution: A

• Using the Distributive Law: (A + B)(A + B') = A(A + B') + B(A + B').
• Simplifying further: A(A + B') = A, and B(A + B') = B' + B = 1.
• Thus, the entire expression simplifies to A.

Question 6: Simplify the Boolean expression A + AB'.

Solution: A + B'

• Using the Distributive Law: A + AB' = A(1 + B').
• Since 1 + B' = 1, the expression simplifies to A + B'.

Question 7: Prove that A + A' = 1.

Solution: 1

• A + A' is an axiom in Boolean algebra known as the Complement Law.
• The sum of a variable and its complement is always equal to 1.

Question 8: Simplify the Boolean expression A'B + AB'.

Solution: A ⊕ B (A XOR B)

• This is the standard expression for XOR (exclusive OR): A ⊕ B = A'B + AB'.

Question 9: Simplify the Boolean expression A' + AB + A'C.

Solution: A' + B + C

• Using the Distributive Law: A' + AB + A'C = A'(1 + C) + AB.
• Since 1 + C = 1, the expression simplifies to A' + AB = A' + B.
• The result is A' + B + C.

Question 10: Simplify the Boolean expression AB + A'B + AB'.

Solution: A + B

• Using the Distributive Law: AB + A'B + AB' = B(A + A') + AB'.
• Since A + A' = 1, the expression simplifies to B + AB' = A + B.

Question 11: Simplify the Boolean expression (A + B)(A + C).

Solution: A + BC

• Using the Distributive Law: (A + B)(A + C) = A(A + C) + B(A + C).
• Simplifying: A + AC + AB + BC = A + AB + BC.
• Using Absorption: A + AB = A, so the expression becomes A + BC.

Question 12: Simplify the Boolean expression A'B + AB.

Solution: B

• Using the Distributive Law: A'B + AB = B(A' + A).
• Since A' + A = 1, the expression simplifies to B.

Question 13: Prove that A + 0 = A.

Solution: A

• This is a Boolean algebra axiom called the Identity Law.
• The OR operation with 0 does not affect the value of A, so A + 0 = A.