---------------------------------------------------------------------- LOGICAL EQUIVALENCE, NEGATION AND INVERSION A TUTORIAL BY JAMES P. DAVIS ---------------------------------------------------------------------- "+" represents a "TRUE" or a "1" or an "ON" state in all cases. "o" represents a "FALSE" or a "0" or an "OFF" state in all cases. "=" represents equivalence: two things being equal to each other. "not" means to negate or change to the opposite state. Parentheses, "(" & ")", are used for grouping operations that should be performed first. Nested parentheses, parentheses within parentheses, means to perform the innermost operation first. ---------------------------------------------------------------------- BASIC LOGICAL TRUTH TABLE ---------------------------------------------------------------------- PROPOSITIONS = PREMISES CASE LOGICAL CONNECTIONS ------------------------------- ------------ ----------------------- 1 2 3 4 ------------------------------- ------------ ----------------------- Basic Premise: P + + o o P = not (not P) Basic Premise: Q + o + o Q = not (not Q) ------------------------------- ------------ ----------------------- Basic Premise: not P o o + + not P = not (P) Basic Premise: not Q o + o + not Q = not (Q) ------------------------------- ------------ ----------------------- A = P and Q + o o o A = not H = not S = Z B = P and (not Q) o + o o B = not G = not T = Y C = (not P) and Q o o + o C = not F = not U = X D = (not P) and (not Q) o o o + D = not E = not V = W ------------------------------- ------------ ----------------------- E = P or Q + + + o E = not D = V = not W F = P or (not Q) + + o + F = not C = U = not X G = (not P) or Q + o + + G = not B = T = not Y H = (not P) or (not Q) o + + + H = not A = S = not Z ------------------------------- ------------ ----------------------- S = not (P and Q) o + + + S = not A = H = not Z T = not (P and (not Q)) + o + + T = not B = G = not Y U = not ((not P) and Q) + + o + U = not C = F = not X V = not ((not P) and (not Q)) + + + o V = not D = E = not W ------------------------------- ------------ ----------------------- W = not (P or Q) o o o + W = D = not E = not V X = not (P or (not Q)) o o + o X = C = not F = not U Y = not ((not P) or Q) o + o o Y = B = not G = not T Z = not ((not P) or (not Q)) + o o o Z = A = not H = not S ------------------------------- ------------ ----------------------- Tautology: P or (not P) + + + + True in all cases. Contradiction: P and (not P) o o o o False in all cases. ------------------------------- ------------ ----------------------- ----------------------------------------------------------------------