Definition:Logical NAND

Definition
NAND (that is, not and), is a binary connective, written symbolically as $p \uparrow q$, whose behaviour is as follows:


 * $p \uparrow q$

is defined as:
 * $p$ and $q$ are not both true.

$p \uparrow q$ is voiced:
 * $p$ nand $q$

The symbol $\uparrow$ is known as the Sheffer stroke, named after Henry Sheffer, who proved an important result about this operation.

Boolean Interpretation
From the above, we see that the boolean interpretations for $\mathbf A \uparrow \mathbf B$ under the model $\mathcal M$ are:


 * $\left({\mathbf A \uparrow \mathbf B}\right)_\mathcal M = \begin{cases}

F & : \mathbf A_\mathcal M = T \text{ and } \mathbf B_\mathcal M = T \\ T & : \text {otherwise} \end{cases}$

Complement
The complement of $\uparrow$ is the conjunction operator.

Truth Function
The NAND connective defines the truth function $f^\uparrow$ as follows:

Truth Table
The truth table of $p \uparrow q$ and its complement is as follows:

$\begin{array}{|cc||c|c|} \hline p & q & p \uparrow q & p \land q \\ \hline F & F & T & F \\ F & T & T & F \\ T & F & T & F \\ T & T & F & T \\ \hline \end{array}$

Notational Variants
Various symbols are encountered that denote the concept of NAND: