The laws of propositional logic help us find logical equivalence between propositions.
What Are De Morgan’s Laws In Logical Propositions?
De Morgan’s laws are found in set theory, computer engineering, and in propositional logic, which is the topic of this post.
Using de Morgan’s laws, we can find equivalency in propositional statements.
According to de Morgan’s laws, the following compound proposition, ¬(T ∨ Y), is logically equivalent to (¬T ∧ ¬Y) and vice-versa. Notice the swapping of the conjunction and disjunction. This is are saying that Not(T or Y) is logically equivalent to Not T and Not Y.
De Morgan’s laws example¬(T ∨ Y) ≡ (¬T ∧ ¬Y) both compound propositions are logically equivalent
De Morgan’s laws example¬(T ∧ Y) ≡ (¬T ∨ ¬Y) this reverses the first law
| De Morgan’s laws: | ¬( p ∨ q ) ≡ ¬p ∧ ¬q | ¬( p ∧ q ) ≡ ¬p ∨ ¬q |
| Idempotent laws: | p ∨ p ≡ p | p ∧ p ≡ p |
| Associative laws: | ( p ∨ q ) ∨ r ≡ p ∨ ( q ∨ r ) | ( p ∧ q ) ∧ r ≡ p ∧ ( q ∧ r ) |
| Commutative laws: | p ∨ q ≡ q ∨ p | p ∧ q ≡ q ∧ p |
| Distributive laws: | p ∨ ( q ∧ r ) ≡ ( p ∨ q ) ∧ ( p ∨ r ) | p ∧ ( q ∨ r ) ≡ ( p ∧ q ) ∨ ( p ∧ r ) |
| Identity laws: | p ∨ F ≡ p | p ∧ T ≡ p |
| Domination laws: | p ∧ F ≡ F | p ∨ T ≡ T |
| Double negation law: | ¬¬p ≡ p | |
| Complement laws: | p ∧ ¬p ≡ F ¬T ≡ F | p ∨ ¬p ≡ T ¬F ≡ T |
| Absorption laws: | p ∨ (p ∧ q) ≡ p | p ∧ (p ∨ q) ≡ p |
| Conditional identities: | p → q ≡ ¬p ∨ q | p ↔ q ≡ ( p → q ) ∧ ( q → p ) |