Binary operations There are 6 possible truth functions of two binary variables : Truth table for all binary logical operators Here is a truth table giving definitions of all 6 of the possible truth functions of two binary variables (P and Q are thus boolean variables: information about notation may be found in Bocheński (959), Enderton (200), and Quine (982); for details about the operators see the Key below): P Q F0 NOR Xq2 ¬p3 ↛4 ¬q5 XOR6 NAND7 AND8 XNOR9 q0 if/then p2 then/if3 OR4 T5 T T F F F F F F F F T T T T T T T T T F F F F F T T T T F F F F T T T T F T F F T T F F T T F F T T F F T T F F F T F T F T F T F T F T F T F T Com ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ L id F F T T T,F T F R id F F T T T,F T F where T = true and F = false. The Com row indicates whether an operator, op, is commutative - P op Q = Q op P. The L id row shows the operator's left identities if it has any - values I such that I op Q = Q. The R id row shows the operator's right identities if it has any - values I such that P op I = P.[note ] The four combinations of input values for p, q, are read by row from the table above. The output function for each p, q combination, can be read, by row, from the table. Key: The key is oriented by column, rather than row. There are four columns rather than four rows, to display the four combinations of p, q, as input. p: T T F F q: T F T F There are 6 rows in this key, one row for each binary function of the two binary variables, p, q. For example, in row 2 of this Key, the value of Converse nonimplication (' ') is solely T, for the column denoted by the unique combination p=F, q=T; while in row 2, the value of that ' ' operation is F for the three remaining columns of p, q. The output row for is thus 2: F F T F and the 6-row[3] key is [3] operator Operation name 0 (F F F F)(p, q) ⊥ false, Opq Contradiction (F F F T)(p, q) NOR p ↓ q, Xpq Logical NOR 2 (F F T F)(p, q) p q, Mpq Converse nonimplication 3 (F F T T)(p, q) ¬p, ~p ¬p, Np, Fpq Negation 4 (F T F F)(p, q) p q, Lpq Material nonimplication 5 (F T F T)(p, q) ¬q, ~q ¬q, Nq, Gpq Negation 6 (F T T F)(p, q) XOR p ⊕ q, Jpq Exclusive disjunction 7 (F T T T)(p, q) NAND p ↑ q, Dpq Logical NAND 8 (T F F F)(p, q) AND p ∧ q, Kpq Logical conjunction 9 (T F F T)(p, q) XNOR p If and only if q, Epq Logical biconditional 0 (T F T F)(p, q) q q, Hpq Projection function (T F T T)(p, q) p q if p then q, Cpq Material implication 2 (T T F F)(p, q) p p, Ipq Projection function 3 (T T F T)(p, q) p q p if q, Bpq Converse implication 4 (T T T F)(p, q) OR p ∨ q, Apq Logical disjunction 5 (T T T T)(p, q) ⊤ true, Vpq Tautology Truth table for most commonly used logical operators Here is a truth table giving definitions of the most commonly used 6 of the 6 possible truth functions of 2 binary variables (P,Q are thus boolean variables): T T T T F T T T T T F F T T F F T F F T F T T F T F F F F F F F T T T T Key: T = true, F = false = AND (logical conjunction) = OR (logical disjunction) = XOR (exclusive or) = XNOR (exclusive nor) = conditional "if-then" = conditional "(then)-if" biconditional or "if-and-only-if" is logically equivalent to : XNOR (exclusive nor).