Truth Table

Build the truth table of a logical expression with intermediate columns, the order of operations, canonical DNF and CNF.

Variables are Latin letters (x, y, z, w, A, B, x1). Operations: ¬ ! not — negation, ∧ & * and — conjunction, ∨ | + or — disjunction, ⊕ ^ xor — exclusive or, → -> — implication, ≡ ↔ == — equivalence. Order: ¬, ∧, ∨ and ⊕, →, ≡; equal operations left to right.

Test yourself

The truth table generator builds the truth table of a logical expression with intermediate columns, shows the order of operations, tells whether the expression is a tautology or a contradiction, and writes the canonical DNF and CNF.

Logical operations

Operation	Symbol	Input	True when
negation (NOT)	¬	! ~ not	the operand is false
conjunction (AND)	∧	& * and	both are true
disjunction (OR)	∨	\| + or	at least one is true
exclusive OR	⊕	^ xor	the values differ
implication	→	->	except 1 → 0
equivalence	≡	== <->	the values are equal

Order: brackets, negation, conjunction, disjunction and XOR, implication, equivalence; equal operations go left to right. With n variables the table has 2ⁿ rows.

Example. (x ∨ y) → ¬z is false only when x ∨ y = 1 and z = 1: in 3 of 8 rows.