connectives

connective priority

¬ not ∧ and ∨ or ⇒ implies ⇔ if and only if

XOR

p XOR q ≡ (p ∨ q) ∧ ¬(p ∧ q)

connectives and logic

just two connectives, ¬ with either ∨ or ∧, are enough to express logical equivalents of all others p ∨ q ≡ ¬(¬p ∧ ¬q) therefore, p xor q ≡ ¬(¬p ∧ ¬q) ∧ ¬(p ∧ q)

NAND

p ↑ q ≡ ¬(p ∧ q)

NOR

p ↓ q ≡ ¬(p ∨ q)

not connectives and logic

just one connective, either nand or nor is enough to express all others ¬p ≡ p ↑ p p ∨ q ≡ ¬p ↑ ¬q and p ∧ q ≡ ¬p ↓ ¬q ¬p ≡ p ↓ p

implies connective

notated p ⇒ q true unless p is true and q is false

converse of an implication

p ⇒ q goes to q ⇒ p equal to the inverse

inverse of an implication

p ⇒ q goes to ¬p ⇒ ¬q equal to the converse

contrapositive of an implication

p ⇒ q goes to ¬q ⇒ ¬p equal to the original implication

if and only if connective

denoted p ⇔ q, can be abbreviated to iff true if both variables have the same truth value, false otherwise

iff expressed as implies

p ⇔ q ≡ (q ⇒ p) ∧ (p ⇒ q)