Advanced Boolean Identities, Consensus, and Duality Quiz

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| By Thames
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Quizzes Created: 7387 | Total Attempts: 9,527,791
| Questions: 15 | Updated: Dec 1, 2025
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1) A XOR A simplifies to?

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.

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About This Quiz
Advanced Boolean Identities, Consensus, And Duality Quiz - Quiz

In this quiz, you’ll explore slightly richer patterns built from the same core laws. You’ll simplify XOR-related expressions (like A ⊕ A and A ⊕ 1), use De Morgan’s theorem in context, recognize equivalence (A IFF B), and apply the consensus theorem and duality. Many questions show longer expressions that... see morecollapse to something simple like A, A OR B, A AND B, or A XOR B once the right law is applied. By the end, you’ll be more confident spotting shortcuts and hidden structures inside complex-looking Boolean expressions. see less

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2) A XOR 1 simplifies to?

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.

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3) A XOR 0 simplifies to?

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.

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

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.

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5) Which of the following is equivalent to (A + B')(A' + C)(B' + 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).
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6) NOT (A AND NOT B) simplifies to?

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.

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

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

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.

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

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

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

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.

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

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

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

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

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.
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A XOR A simplifies to?
A XOR 1 simplifies to?
A XOR 0 simplifies to?
A OR (B AND C) = (A OR B) AND (A OR C) represents which law?
Which of the following is equivalent to (A + B')(A' + C)(B' + C)?
NOT (A AND NOT B) simplifies to?
(A AND B) OR (NOT A AND B) OR (A AND NOT B) simplifies to?
A OR (NOT A AND B) simplifies to?
(A OR B) AND (NOT A OR NOT B) simplifies to?
Dual of A OR A AND B = A is?
NOT (NOT A OR NOT B) simplifies to?
(A OR A AND B) OR A simplifies to?
(A OR B) AND (A OR NOT B) OR NOT A simplifies to?
(A OR B) AND (NOT A OR B) simplifies to?
A AND B OR A AND NOT B OR NOT A AND B simplifies to?
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