Simplification Rules Quiz: Reduce Logical Expressions Clearly

Identity law: X ∧ 1 = X because AND with true leaves the value unchanged; if X = 1 then 1 ∧ 1 = 1, and if X = 0 then 0 ∧ 1 = 0, so the output always equals X.

Identity law: X ∨ 0 = X because OR with false cannot make the expression true; if X = 1 then 1 ∨ 0 = 1, and if X = 0 then 0 ∨ 0 = 0, so the result is exactly X.

Annihilator law: X ∧ 0 = 0 because AND requires both inputs true, and the 0 (false) input forces the entire expression to be false regardless of X.

Annihilator law: X ∨ 1 = 1 because OR requires only one input to be true, and since 1 is always true, the whole expression becomes true no matter what X is.

Complement law: X ∨ ¬X = 1 because a variable and its negation exhaust every possibility; one of them must be true, so the OR is always true.

Complement law: X ∧ ¬X = 0 because a variable and its negation can never both be true simultaneously; the AND therefore always evaluates to false.

Absorption law: A ∨ (A ∧ B) = A because if A is true the whole expression is true already, and if A is false then A ∧ B is also false, so the expression always simplifies to A.

Absorption law: A ∧ (A ∨ B) = A because if A is false the entire AND becomes false, and if A is true then A ∨ B is true, so the result depends entirely on A.

Factor out A: (A ∧ B) ∨ (A ∧ C) = A ∧ (B ∨ C) because both terms require A; once A is factored, the remaining condition is whether B or C is true.

Distributive law: (A ∨ B) ∧ (A ∨ C) = A ∨ (B ∧ C) because OR distributes over AND in Boolean algebra; the expression is true either when A is true or when both B and C are true.

Factor B: (A ∧ B) ∨ (A' ∧ B) = B because factoring yields B ∧ (A ∨ A'), and since A ∨ ¬A = 1, the expression becomes B ∧ 1 = B.

Factor A ∧ B: (A ∧ B ∧ C) ∨ (A ∧ B ∧ C') = A ∧ B because factoring gives A ∧ B ∧ (C ∨ ¬C), and since C ∨ ¬C = 1, the entire expression reduces to A ∧ B.

Distribute: A ∧ (¬A ∨ B) = (A ∧ ¬A) ∨ (A ∧ B) = A ∧ B because A ∧ ¬A = 0 by the complement law, leaving only A ∧ B.

De Morgan: ¬(A ∧ B) = ¬A ∨ ¬B because negating an AND means at least one component must be false, which is captured by OR of the negations.

De Morgan: ¬(A ∨ B) = ¬A ∧ ¬B because negating an OR means both components must be false, which is expressed by AND of the negations.

B ∨ ¬B = 1, so (A ∧ B) ∨ (A ∧ B') simplifies to A because factoring gives A ∧ (B ∨ ¬B) = A ∧ 1 = A.

Distributive law: A ∨ (B ∧ C) = (A ∨ B) ∧ (A ∨ C) because OR distributes over AND; the expression is true if A is true or if both B and C are true.

Since B ∨ ¬B = 1, A ∧ (B ∨ ¬B) = A because the expression becomes A ∧ 1, and AND with 1 leaves the value unchanged.

Absorption law: (A ∨ B) ∧ A = A because if A is true the expression is true, and if A is false the AND is false, so the value is determined solely by A.

Consensus simplification: A ∨ (¬A ∧ B) = A ∨ B because if A is true the whole expression is true, and if A is false it reduces to B, so the expression is true exactly when either A or B is true.