Truth tables
AND, OR, NOT, XOR, implication and biconditional: full table with intermediate columns and equivalence checking between two formulas.
Syntax: single-letter variables (p, q, r…); NOT ¬ !, AND ∧ &, OR ∨ |, XOR ⊕ ^, implies -> →, iff <-> ↔. Words work too: p and q, not p. Note: the letters e and o act as word operators (AND/OR) and can't be used as variables.
Reading the table
Each row is one combination of variable values (T/F); intermediate columns show the formula's pieces, so you see WHERE the value flips. The highlighted last column is the whole formula: all T = tautology, all F = contradiction.
Implication, the weird one
p → q is false only when p is true and q false: “if it rains I take an umbrella” isn't disproved on sunny days. The equivalence p → q ≡ ¬p ∨ q holds — try it in the comparison field: identical columns.
Why equivalence matters
De Morgan's laws — ¬(p ∧ q) ≡ ¬p ∨ ¬q — are the bread and butter of circuit simplification and programming conditions: equivalent formulas can always swap. The tool's comparison checks equivalence over every possible combination.
Note
Up to 5 variables (32 rows), the most that stays readable on a page. Standard textbook precedence; when in doubt, parentheses always win.