Equivalence Propositional Laws
☰Fullscreen
Table of Content:
1. Identity Laws
| Law | Expression |
|---|---|
| Identity (AND) | P ∧ T ≡ P |
| Identity (OR) | P ∨ F ≡ P |
2. Domination Laws
| Law | Expression |
|---|---|
| Domination (OR) | P ∨ T ≡ T |
| Domination (AND) | P ∧ F ≡ F |
3. Idempotent Laws
| Law | Expression |
|---|---|
| Idempotent (OR) | P ∨ P ≡ P |
| Idempotent (AND) | P ∧ P ≡ P |
4. Double Negation Law
| Law | Expression |
|---|---|
| Double Negation | ¬(¬P) ≡ P |
5. Commutative Laws
| Law | Expression |
|---|---|
| Commutative (OR) | P ∨ Q ≡ Q ∨ P |
| Commutative (AND) | P ∧ Q ≡ Q ∧ P |
6. Associative Laws
| Law | Expression |
|---|---|
| Associative (OR) | (P ∨ Q) ∨ R ≡ P ∨ (Q ∨ R) |
| Associative (AND) | (P ∧ Q) ∧ R ≡ P ∧ (Q ∧ R) |
7. Distributive Laws
| Law | Expression |
|---|---|
| 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
| Law | Expression |
|---|---|
| De Morgan (OR) | ¬(P ∨ Q) ≡ ¬P ∧ ¬Q |
| De Morgan (AND) | ¬(P ∧ Q) ≡ ¬P ∨ ¬Q |
9. Absorption Laws
| Law | Expression |
|---|---|
| Absorption (OR) | P ∨ (P ∧ Q) ≡ P |
| Absorption (AND) | P ∧ (P ∨ Q) ≡ P |
10. Negation Laws
| Law | Expression |
|---|---|
| Negation (OR) | P ∨ ¬P ≡ T |
| Negation (AND) | P ∧ ¬P ≡ F |
11. Implication Law
| Law | Expression |
|---|---|
| Implication | P → Q ≡ ¬P ∨ Q |
12. Biconditional Law
| Law | Expression |
|---|---|
| Biconditional | P ↔ Q ≡ (P → Q) ∧ (Q → P) |
13. Contrapositive Law
| Law | Expression |
|---|---|
| Contrapositive | P → Q ≡ ¬Q → ¬P |
14. Alternative Ways to Write the Same Equivalences
Logical Operator Variations
| Standard Form | Alternative Forms |
|---|---|
| P ∧ Q | P AND Q, P · Q |
| P ∨ Q | P OR Q, P + Q |
| ¬P | NOT P, P' |
| P → Q | IF P THEN Q, ¬P ∨ Q |
| P ↔ Q | P iff Q, (P → Q) ∧ (Q → P) |
Alternative Equivalent Expressions
| Law | Equivalent Forms |
|---|---|
| Identity | P ∧ T ≡ T ∧ P, P ∨ F ≡ F ∨ P |
| Domination | P ∨ T ≡ T, T ∨ P ≡ T |
| Idempotent | P ∧ P ≡ P, P ∨ P ≡ P |
| Double Negation | ¬¬P ≡ P |
| Commutative | P ∧ Q ≡ Q ∧ P, P ∨ Q ≡ Q ∨ P |
| Associative | P ∧ (Q ∧ R) ≡ (P ∧ Q) ∧ R |
| Distributive | (P ∨ Q) ∧ R ≡ (P ∧ R) ∨ (Q ∧ R) |
| De Morgan | ¬(P ∨ Q) ≡ ¬P ∧ ¬Q, ¬(PQ) ≡ P' + Q' |
| Absorption | P + PQ ≡ P, P(P + Q) ≡ P |
| Negation | P + P' ≡ T, PP' ≡ F |
| Implication | P → Q ≡ ¬P ∨ Q ≡ P' + Q |
| Biconditional | P ↔ Q ≡ (P ∧ Q) ∨ (¬P ∧ ¬Q) |
| Contrapositive | P → Q ≡ ¬Q → ¬P |