NAND, also known as the Sheffer stroke, is a connective in logic equivalent to the composition NOT AND that yields true if any condition
is false, and false if all conditions
are true. 
NAND 
is equivalent to 
,
where 
denotes NOT
and 
denotes AND.
In propositional calculus, the term alternative denial is used to refer to the NAND
connective. Notations for NAND include 
and 
(Mendelson 1997, p. 26). The NAND operation is implemented as Nand[A,
B, ...]. The circuit diagram symbol for a NAND gate is illustrated above.
The NAND binary operator has the following truth table (Mendelson 1997, p. 27).
 |  |  |
| T | T | F |
| T | F | T |
| F | T | T |
| F | F | T |
NAND gates implement the NAND operation in digital circuits. Sheffer (1913) gave an axiom system for Boolean algebra based on a single binary operator. Whitehead and Russell (1927) used NAND as a foundation for propositional calculus.
The dual operation to NAND is NOR, also called Peirce's arrow (Zach 2016). Complementing both inputs and the output of NOR gives NAND,
 |
(1)
|
The two operations have different truth tables, but each is functionally complete. In particular, NOT, AND, and OR can all be expressed using only NAND,
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
Consequently, every Boolean function of one or more variables can be expressed using only NAND.
The EML operator gives an analogous single binary operation for constructing a specified scientific-calculator basis of elementary functions and operations.
