Compound Propositions

p ↔ q equivalence

(p → q) ∧ (q → p)

Simplify: (p → r) ∨ (q → r)

(p ∧ q) → r

Simplify: (p → r) ∧ (q → r)

(p ∨ q) → r

Equivalence: Associative laws

(p ∨ q) ∨ r ≡ p ∨ (q ∨ r) (p ∧ q) ∧ r ≡ p ∧ (q ∧ r)

Define contradiction.

A compound proposition that is always false p ∧¬p

Define tautology

A compound proposition that is always true, no matter what the truth values of the propositional variables that occur in it p ∨¬p

Define contingency

A compound proposition that is neither a tautology nor a contradiction

Simplify: (p → q) ∧ (p → r)

p → (q ∧ r)

Simplify: (p → q) ∨ (p → r)

p → (q ∨ r)

Simplify ¬p ∨ q

p → q

Simplify ¬q →¬p

p → q

Simplify: (p → q) ∧ (q → p)

p ↔ q

Simplify: (p ∧ q) ∨ (¬p ∧¬q)

p ↔ q

Simplify: ¬p ↔¬q

p ↔ q

Simplify: ¬(p ↔ q)

p ↔¬q

Equivalence:Identity laws

p ∧ T ≡ p p ∨ F ≡ p

Simplify: ¬(p →¬q)

p ∧ q

Simplify: ¬(p → q)

p ∧¬q

Equivalence:Absorption laws

p ∨ (p ∧ q) ≡ p p ∧ (p ∨ q) ≡ p

Equivalence: Distributive laws

p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r) p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)

Equivalence: Domination laws

p ∨ T ≡ T p ∧ F ≡ F

Equivalence: Idempotent laws

p ∨ p ≡ p p ∧ p ≡ p

Simplify: ¬p → q

p ∨ q

Equivalence Commutative laws

p ∨ q ≡ q ∨ p p ∧ q ≡ q ∧ p

Equivalence: Negation laws

p ∨¬p ≡ T p ∧¬p ≡ F

Let p and q be compound propositions. Define p ≡ q

two compound propositions always have the same truth value p and q are logically equivalent, p ↔ q is a tautology..

De Morgan's Laws

¬(p ∧ q) ≡ ¬p ∨¬q ¬(p ∨ q) ≡ ¬p ∧¬q

Equivalence: Double negation law

¬(¬p) ≡ p