Advanced Boolean Simplification & Consensus Quiz (Undergraduate Level)

1) A XOR A simplifies to?

A

1

0

NOT A

The XOR operation (exclusive OR) is defined as A XOR B = (A AND NOT B) OR (NOT A AND B). For A XOR A, this becomes (A AND NOT A) OR (NOT A AND A) = (0) OR (0) = 0, since A AND NOT A is always false.

Explanation

The XOR operation (exclusive OR) is defined as A XOR B = (A AND NOT B) OR (NOT A AND B). For A XOR A, this becomes (A AND NOT A) OR (NOT A AND A) = (0) OR (0) = 0, since A AND NOT A is always false.

2) A XOR 1 simplifies to?

NOT A

A

1

0

A XOR 1 = (A AND NOT 1) OR (NOT A AND 1). Since NOT 1 is 0, this becomes (A AND 0) OR (NOT A AND 1) = 0 OR NOT A = NOT A. Thus, XOR with 1 flips the value of A.

Explanation

A XOR 1 = (A AND NOT 1) OR (NOT A AND 1). Since NOT 1 is 0, this becomes (A AND 0) OR (NOT A AND 1) = 0 OR NOT A = NOT A. Thus, XOR with 1 flips the value of A.

3) A XOR 0 simplifies to?

A

NOT A

0

1

A XOR 0 = (A AND NOT 0) OR (NOT A AND 0). Since NOT 0 is 1, this becomes (A AND 1) OR (NOT A AND 0) = A OR 0 = A. Thus, XOR with 0 leaves A unchanged.

Explanation

A XOR 0 = (A AND NOT 0) OR (NOT A AND 0). Since NOT 0 is 1, this becomes (A AND 1) OR (NOT A AND 0) = A OR 0 = A. Thus, XOR with 0 leaves A unchanged.

4) A OR (B AND C) = (A OR B) AND (A OR C) represents which law?

Absorption

Inverse Distributive

Consensus

Distributive

This equation represents the distributive law of OR over AND. It states that OR can be distributed over AND, similar to how multiplication distributes over addition in arithmetic.

Explanation

This equation represents the distributive law of OR over AND. It states that OR can be distributed over AND, similar to how multiplication distributes over addition in arithmetic.

5) Which of the following is equivalent to (A + B')(A' + C)(B' + C)?

(A + B')(A' + C)

(A + B')(B' + C)

(A' + C)(B' + C)

A + B' + C

This is a direct application of the consensus theorem in its product-of-sums form. The consensus theorem states that (X+Y)(X'+Z)(Y+Z) = (X+Y)(X'+Z), where (Y+Z) is the consensus term of the first two factors. In this case, with X=A, Y=B', and Z=C, the term (B'+ C) is the consensus term of (A+B') and (A'+C). Since the consensus term is redundant when the other two terms are present, we can eliminate (B'+ C), simplifying to just (A+ B')(A'+ C).

Explanation

This is a direct application of the consensus theorem in its product-of-sums form. The consensus theorem states that (X+Y)(X'+Z)(Y+Z) = (X+Y)(X'+Z), where (Y+Z) is the consensus term of the first two factors. In this case, with X=A, Y=B', and Z=C, the term (B'+ C) is the consensus term of (A+B') and (A'+C). Since the consensus term is redundant when the other two terms are present, we can eliminate (B'+ C), simplifying to just (A+ B')(A'+ C).

6) NOT (A AND NOT B) simplifies to?

NOT A OR B

NOT A AND B

A OR B

A AND NOT B

Using De Morgan’s theorem, NOT (A AND NOT B)=NOT A OR NOT NOT B. Since NOT NOT B is B, the simplification results in NOT A OR B.

Explanation

Using De Morgan’s theorem, NOT (A AND NOT B)=NOT A OR NOT NOT B. Since NOT NOT B is B, the simplification results in NOT A OR B.

7) (A AND B) OR (NOT A AND B) OR (A AND NOT B) simplifies to?

A XOR B

A AND B

A OR B

1

This expression covers all cases where at least one of A or B is true: (A AND B) is both true, (NOT A AND B) is only B true, and (A AND NOT B) is only A true. Thus, the overall expression is true when A is true or B is true, which is A OR B.

Explanation

This expression covers all cases where at least one of A or B is true: (A AND B) is both true, (NOT A AND B) is only B true, and (A AND NOT B) is only A true. Thus, the overall expression is true when A is true or B is true, which is A OR B.

8) A OR (NOT A AND B) simplifies to?

A

B

A AND B

A OR B

Using the distributive law, A OR (NOT A AND B)=(A OR NOT A) AND (A OR B)=1 AND (A OR B)=A OR B.

Explanation

Using the distributive law, A OR (NOT A AND B)=(A OR NOT A) AND (A OR B)=1 AND (A OR B)=A OR B.

9) (A OR B) AND (NOT A OR NOT B) simplifies to?

A XOR B

NOT (A AND B)

A AND B

NOT (A OR B)

This expression represents the exclusive OR (XOR) of A and B. It is true when A and B are different: (A OR B) is true if at least one is true, and (NOT A OR NOT B) is true if not both are true, so together, it is true when exactly one is true.

Explanation

This expression represents the exclusive OR (XOR) of A and B. It is true when A and B are different: (A OR B) is true if at least one is true, and (NOT A OR NOT B) is true if not both are true, so together, it is true when exactly one is true.

10) Dual of A OR A AND B = A is?

A AND A OR B = A

A AND (A OR B) = A

A OR (A AND B) = A

NOT A AND NOT B = NOT A

The dual of a Boolean equation is obtained by swapping OR with AND and 1 with 0 (if present). The dual of A OR (A AND B) = A is A AND (A OR B) = A, which is also an absorption law.

Explanation

The dual of a Boolean equation is obtained by swapping OR with AND and 1 with 0 (if present). The dual of A OR (A AND B) = A is A AND (A OR B) = A, which is also an absorption law.

11) NOT (NOT A OR NOT B) simplifies to?

NOT A OR B

A OR B

A AND B

NOT A AND B

Using De Morgan’s theorem, NOT (NOT A OR NOT B) = NOT NOT A AND NOT NOT B. Since NOT NOT A is A and NOT NOT B is B, the simplification results in A AND B.

Explanation

Using De Morgan’s theorem, NOT (NOT A OR NOT B) = NOT NOT A AND NOT NOT B. Since NOT NOT A is A and NOT NOT B is B, the simplification results in A AND B.

12) (A OR A AND B) OR A simplifies to?

A

A OR B

B

1

First, A OR (A AND B) simplifies to A by the absorption law. Then, A OR A simplifies to A by the idempotent law. Thus, the entire expression simplifies to A.

Explanation

First, A OR (A AND B) simplifies to A by the absorption law. Then, A OR A simplifies to A by the idempotent law. Thus, the entire expression simplifies to A.

13) (A OR B) AND (A OR NOT B) OR NOT A simplifies to?

1

A OR NOT A

A OR NOT A AND B

NOT A OR A

First, (A OR B) AND (A OR NOT B) simplifies to A by the distributive law as shown in previous questions. Then, A OR NOT A simplifies to 1 by the complement law. Thus, the expression becomes 1.

Explanation

First, (A OR B) AND (A OR NOT B) simplifies to A by the distributive law as shown in previous questions. Then, A OR NOT A simplifies to 1 by the complement law. Thus, the expression becomes 1.

14) (A OR B) AND (NOT A OR B) simplifies to?

NOT B

B

A AND B

A OR B

Using the distributive law, (A OR B) AND (NOT A OR B) = (A AND NOT A) OR (A AND B) OR (B AND NOT A) OR (B AND B) = 0 OR (A AND B) OR (NOT A AND B) OR B. This simplifies to B OR (B AND (A OR NOT A)) = B OR (B AND 1) = B OR B = B.

Explanation

Using the distributive law, (A OR B) AND (NOT A OR B) = (A AND NOT A) OR (A AND B) OR (B AND NOT A) OR (B AND B) = 0 OR (A AND B) OR (NOT A AND B) OR B. This simplifies to B OR (B AND (A OR NOT A)) = B OR (B AND 1) = B OR B = B.

15) A AND B OR A AND NOT B OR NOT A AND B simplifies to?

1

A AND B

A OR B

A XOR B

This expression covers all cases where A or B is true. Specifically, A AND B covers both true, A AND NOT B covers only A true, and NOT A AND B covers only B true. Thus, it simplifies to A OR B.

Explanation

This expression covers all cases where A or B is true. Specifically, A AND B covers both true, A AND NOT B covers only A true, and NOT A AND B covers only B true. Thus, it simplifies to A OR B.

Advanced Boolean Identities, Consensus, and Duality Quiz