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).