Truth Function - Table of Binary Truth Functions

Table of Binary Truth Functions

There are sixteen Boolean functions associating the inputs P and Q with four-digit binary outputs. Any of these functions is identified as a truth table of a certain logical connective in classical logic, including several degenerate cases such as a function not depending on one or both of its arguments.

Contradiction/False
Notation Equivalent
formulas		 Truth table Venn diagram

"bottom"		 P ¬P
Opq
Q
0 1
P 0 0 0
1 0 0


Tautology/True
Notation Equivalent
formulas		 Truth table Venn diagram

"top"		 P ¬P
Vpq
Q
0 1
P 0 1 1
1 1 1


Proposition P
Notation Equivalent
formulas		 Truth table Venn diagram
P p
Ipq
Q
0 1
P 0 0 0
1 1 1


Negation of P
Notation Equivalent
formulas		 Truth table Venn diagram
¬P
~P		 Np
Fpq
Q
0 1
P 0 1 1
1 0 0


Proposition Q
Notation Equivalent
formulas		 Truth table Venn diagram
Q q
Hpq
Q
0 1
P 0 0 1
1 0 1


Negation of Q
Notation Equivalent
formulas		 Truth table Venn diagram
¬Q
~Q		 Nq
Gpq
Q
0 1
P 0 1 0
1 1 0


Conjunction
Notation Equivalent
formulas		 Truth table Venn diagram
P Q
P & Q
P · Q
P AND Q		 P ¬Q
¬P Q
¬P ¬Q
Kpq
Q
0 1
P 0 0 0
1 0 1


Alternative denial
Notation Equivalent
formulas		 Truth table Venn diagram
P ↑ Q
P | Q
P NAND Q		 P → ¬Q
¬P ← Q
¬P ¬Q
Dpq
Q
0 1
P 0 1 1
1 1 0


Disjunction
Notation Equivalent
formulas		 Truth table Venn diagram
P Q
P OR Q		 P ¬Q
¬P → Q
¬P ↑ ¬Q
¬(¬P ¬Q)
Apq
Q
0 1
P 0 0 1
1 1 1


Joint denial
Notation Equivalent
formulas		 Truth table Venn diagram
P ↓ Q
P NOR Q		 P ¬Q
¬P Q
¬P ¬Q
Xpq
Q
0 1
P 0 1 0
1 0 0


Material nonimplication
Notation Equivalent
formulas		 Truth table Venn diagram
P Q
P Q		 P ¬Q
¬P ↓ Q
¬P ¬Q
Lpq
Q
0 1
P 0 0 0
1 1 0


Material implication
Notation Equivalent
formulas		 Truth table Venn diagram
P → Q
P Q		 P ↑ ¬Q
¬P Q
¬P ← ¬Q
Cpq
Q
0 1
P 0 1 1
1 0 1


Converse nonimplication
Notation Equivalent
formulas		 Truth table Venn diagram
P Q
P Q		 P ↓ ¬Q
¬P Q
¬P ¬Q
Mpq
Q
0 1
P 0 0 1
1 0 0


Converse implication
Notation Equivalent
formulas		 Truth table Venn diagram
P Q
P Q		 P ¬Q
¬P ↑ Q
¬P → ¬Q
Bpq
Q
0 1
P 0 1 0
1 1 1


Exclusive disjunction
Notation Equivalent
formulas		 Truth table Venn diagram
P Q
P Q
P Q
P XOR Q		 P ¬Q
¬P Q
¬P ¬Q
Jpq
Q
0 1
P 0 0 1
1 1 0


Biconditional
Notation Equivalent
formulas		 Truth table Venn diagram
P Q
P ≡ Q
P XNOR Q
P IFF Q		 P ¬Q
¬P Q
¬P ¬Q
Epq
Q
0 1
P 0 1 0
1 0 1


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