Table of Contents

    Equivalence Propositional Laws

    Rumman Ansari April 18, 2026 106 views

    1. Identity Laws

    LawExpression
    Identity (AND)P ∧ T ≡ P
    Identity (OR)P ∨ F ≡ P

    2. Domination Laws

    LawExpression
    Domination (OR)P ∨ T ≡ T
    Domination (AND)P ∧ F ≡ F

    3. Idempotent Laws

    LawExpression
    Idempotent (OR)P ∨ P ≡ P
    Idempotent (AND)P ∧ P ≡ P

    4. Double Negation Law

    LawExpression
    Double Negation¬(¬P) ≡ P

    5. Commutative Laws

    LawExpression
    Commutative (OR)P ∨ Q ≡ Q ∨ P
    Commutative (AND)P ∧ Q ≡ Q ∧ P

    6. Associative Laws

    LawExpression
    Associative (OR)(P ∨ Q) ∨ R ≡ P ∨ (Q ∨ R)
    Associative (AND)(P ∧ Q) ∧ R ≡ P ∧ (Q ∧ R)

    7. Distributive Laws

    LawExpression
    Distributive (AND over OR)P ∧ (Q ∨ R) ≡ (P ∧ Q) ∨ (P ∧ R)
    Distributive (OR over AND)P ∨ (Q ∧ R) ≡ (P ∨ Q) ∧ (P ∨ R)

    8. De Morgan’s Laws

    LawExpression
    De Morgan (OR)¬(P ∨ Q) ≡ ¬P ∧ ¬Q
    De Morgan (AND)¬(P ∧ Q) ≡ ¬P ∨ ¬Q

    9. Absorption Laws

    LawExpression
    Absorption (OR)P ∨ (P ∧ Q) ≡ P
    Absorption (AND)P ∧ (P ∨ Q) ≡ P

    10. Negation Laws

    LawExpression
    Negation (OR)P ∨ ¬P ≡ T
    Negation (AND)P ∧ ¬P ≡ F

    11. Implication Law

    LawExpression
    ImplicationP → Q ≡ ¬P ∨ Q

    12. Biconditional Law

    LawExpression
    BiconditionalP ↔ Q ≡ (P → Q) ∧ (Q → P)

    13. Contrapositive Law

    LawExpression
    ContrapositiveP → Q ≡ ¬Q → ¬P

    14. Alternative Ways to Write the Same Equivalences

    Logical Operator Variations

    Standard FormAlternative Forms
    P ∧ QP AND Q, P · Q
    P ∨ QP OR Q, P + Q
    ¬PNOT P, P'
    P → QIF P THEN Q, ¬P ∨ Q
    P ↔ QP iff Q, (P → Q) ∧ (Q → P)

    Alternative Equivalent Expressions

    LawEquivalent Forms
    IdentityP ∧ T ≡ T ∧ P, P ∨ F ≡ F ∨ P
    DominationP ∨ T ≡ T, T ∨ P ≡ T
    IdempotentP ∧ P ≡ P, P ∨ P ≡ P
    Double Negation¬¬P ≡ P
    CommutativeP ∧ Q ≡ Q ∧ P, P ∨ Q ≡ Q ∨ P
    AssociativeP ∧ (Q ∧ R) ≡ (P ∧ Q) ∧ R
    Distributive(P ∨ Q) ∧ R ≡ (P ∧ R) ∨ (Q ∧ R)
    De Morgan¬(P ∨ Q) ≡ ¬P ∧ ¬Q, ¬(PQ) ≡ P' + Q'
    AbsorptionP + PQ ≡ P, P(P + Q) ≡ P
    NegationP + P' ≡ T, PP' ≡ F
    ImplicationP → Q ≡ ¬P ∨ Q ≡ P' + Q
    BiconditionalP ↔ Q ≡ (P ∧ Q) ∨ (¬P ∧ ¬Q)
    ContrapositiveP → Q ≡ ¬Q → ¬P

    Related Questions

    Question 1

    Prove that \( X + X' \) is a tautology and \( X \cdot X' \) is a contradiction.
    Question 2

    Prove that \( X + 1 \) is a tautology.
    Question 3

    Consider the following simple propositions:

    • \( A \): It is raining.
    • \( B \): Wind is blowing.
    • \( C \): I am not driving.

    From these, create the following compound proportions:

    (i) \( A \lor B \)    (ii) \( \sim B \)    (iii) \( \sim A \cdot C \)    (iv) \( A \cdot \sim C \)    (v) \( A + B \cdot C \)

    Question 4

    Prove that \( p \Leftrightarrow q = (p \Rightarrow q) \cdot (q \Rightarrow p) \).
    Question 5

    Prove that \( p \Leftrightarrow q = q \Leftrightarrow p \).
    Question 6

    Prove that \( p \Rightarrow q = \bar{p} + q \).
    Question 7

    Using a truth table, prove that \( p \Rightarrow q \) is equivalent to \( \sim q \Rightarrow \sim p \).
    Question 8

    Construct a truth table for the expression \( (A \cdot (A + B)) \). What single term is the expression equivalent to?