Boolean Laws Quiz: Apply Algebraic Logic Rules

XOR is true only when the inputs differ. When both inputs are A, they are always the same, so A XOR A is always false, which simplifies to 0. Since XOR measures “difference,” using the same value twice guarantees equality, making the expression false for both truth values of A. This mirrors how subtraction of identical bits yields zero in digital logic.

XOR with 1 flips the value: when A = 0, the result is 1; when A = 1, the result is 0. Therefore A XOR 1 is equivalent to NOT A. Here, 1 acts as a toggling operator, reversing the bit every time. This is the basis of bit-flipping operations in computer architecture, where XOR with 1 inverts a binary value.

XOR with 0 leaves the input unchanged: 0 XOR 0 = 0 and 1 XOR 0 = 1, so A XOR 0 simplifies directly to A. The bit 0 is the identity element for XOR because it does not introduce any difference between the inputs. This property is why XOR is used for reversible encoding.

This is the distributive law where OR distributes over AND: A ∨ (B ∧ C) is rewritten as (A ∨ B) ∧ (A ∨ C), similar to algebraic distribution. This transformation preserves logical equivalence by expanding the expression into separate OR clauses, each capturing one branch of the inner conjunction—exactly like expanding products over sums in arithmetic.

Apply De Morgan's law: NOT(A ∧ ¬B) = NOT A OR NOT(¬B) = ¬A ∨ B. This removes the inner AND and flips both literals and the connective. This law reverses the structure of the statement by transforming the conjunction into a disjunction and simultaneously negating each component, cleaning up nested negations and making the expression easier to evaluate.

Use De Morgan's law on OR: ¬(¬A ∨ ¬B) = ¬(¬A) ∧ ¬(¬B) = A ∧ B. Double negation returns A and B to their original form. Negating a disjunction forces both components to be false, which is exactly the same requirement as asserting both A and B are true. This reversal forms the backbone of contraposition and many logical proofs.

Factor A using distribution: (A ∧ B) ∨ (A ∧ ¬B) = A ∧ (B ∨ ¬B). Since B ∨ ¬B = 1, the result is A ∧ 1 = A. By pulling out the common factor A, the expression collapses into a tautology inside the parentheses. Because B ∨ ¬B covers all possibilities, only A remains relevant to the truth of the entire expression.

The expression is true exactly when A and B have the same value (both 0 or both 1). That is the definition of equivalence, so it simplifies to A IFF B. A XNOR pattern arises here: the expression matches the behavior of equality between truth values, capturing the logic that the two statements must agree for the whole statement to be true.

Use distributive or consensus reasoning: A ∨ (¬A ∧ B) = (A ∨ ¬A) ∧ (A ∨ B) = 1 ∧ (A ∨ B) = A ∨ B. The expression simplifies because the term A ∨ ¬A is always true, eliminating one branch. This is the absorption principle, where A already guarantees the result regardless of the extra condition.

Expand or recognize the pattern: (A∨B)∧(A∨¬B) = A ∨ (B ∧ ¬B) = A ∨ 0 = A. The conflicting B and NOT B cancel out, leaving A. The inner contradiction B ∧ ¬B is always false, making its contribution irrelevant. The OR with A then becomes the dominating term.

Distribute and simplify: (A∨B)∧(¬A∨B) = B ∨ (A ∧ ¬A) = B ∨ 0 = B. Having B in both clauses forces the result to depend only on B. Both expressions provide conditions under which B alone is sufficient to make the statement true, while the contradictory A ∧ ¬A eliminates other branches.

By the consensus theorem, the term (¬B ∨ C) is redundant: (A∨¬B)(¬A∨C)(¬B∨C) simplifies to (A∨¬B)(¬A∨C) while preserving the same truth table. Consensus identifies expressions that provide no additional logical coverage. Removing such terms simplifies the formula without affecting its behavior.

To form the dual, swap OR and AND (and 0 with 1 when present). The dual of A ∨ (A ∧ B) = A is A ∧ (A ∨ B) = A. Duality transforms logical expressions while preserving structural symmetry. The resulting statement highlights the mirror identity involving AND instead of OR.

De Morgan's law for OR states that negating a disjunction converts it into a conjunction of negated terms: ¬(A∨B)=¬A∧¬B. The logic is that for the OR to be false, both pieces must fail—hence the AND of negated components correctly captures the condition.

De Morgan's law for AND states that negating a conjunction yields a disjunction of negations: ¬(A∧B)=¬A∨¬B. A conjunction is only false when at least one input is false, so the negation becomes an OR of the components being false.

The OR-identity is 0: adding 0 by OR does not change the value, so A ∨ 0 = A is the identity law for OR. The value 0 acts as the neutral element, leaving A unaffected, just as adding zero changes nothing in arithmetic.

The AND-identity is 1: multiplying by 1 in Boolean algebra leaves the value unchanged, so A ∧ 1 = A is the identity law for AND. Here, 1 functions as the multiplicative identity of Boolean logic, because any statement AND true reproduces the original statement.

The commutative law for OR states that the order of operands does not matter: A ∨ B has the same value as B ∨ A for all inputs. Like ordinary addition, order does not influence outcomes. This property lets expressions be rearranged freely to simplify or compare them.

The associative law for OR says that grouping of OR terms does not affect the result: (A∨B)∨C = A∨(B∨C). This ensures that OR operations can be combined without considering parentheses, enabling simplification of longer expressions.

First combine the first two terms: (A∧B)∨(A∧¬B)=A. Then A ∨ (¬A∧B) simplifies further to A ∨ B, using the same absorption/consensus reasoning as in A ∨ (¬A∧B). The initial simplification extracts A as a common factor, and the remaining expression again collapses through logical absorption, leaving the simpler OR of A and B.