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Setting Standards

Standard Operator Functions

Among the many strange and complex truth functions that are possible when the concept is generalized, there are several very boring functions that only ever return the value of one operand. These standard operator functions will return zero when their specific operand is zero and one when that operand is one. A standard operator is denoted $\sigma^n_i$, where $n$ is the number of operands in the truth function and $i$ is the specific operand represented by the standard operator. In general the standard operator is defined as:

$$ \left.\left(\textbf{A}^n\right)_{\sigma^n_i}\right|_{\left(\textbf{A}^n\right)_\text{CV}=j}=A_{i-1}\ \forall j\in\left[0,2^n-1\right]\ni i\in\left[1,n\right] $$ Common Standard Operators

Two Operand Standard Operators

$$ \begin{array}{c|c||c} A_0 & A_1 & \left(\textbf{A}^2\right)_{\sigma^2_1} \\ \hline 0 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 1 & 1 \end{array} $$ $$ \begin{array}{c|c||c} A_0 & A_1 & \left(\textbf{A}^2\right)_{\sigma^2_2} \\ \hline 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 1 & 1 \end{array} $$
$\sigma^2_1=\mathtt{0xA}$ $\sigma^2_2=\mathtt{0xC}$

Three Operand Standard Operators

$$ \begin{array}{c|c|c||c} A_0 & A_1 & A_2 & \left(\textbf{A}^3\right)_{\sigma^3_1} \\ \hline 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 \end{array} $$ $$ \begin{array}{c|c|c||c} A_0 & A_1 & A_2 & \left(\textbf{A}^3\right)_{\sigma^3_2} \\ \hline 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \end{array} $$ $$ \begin{array}{c|c|c||c} A_0 & A_1 & A_2 & \left(\textbf{A}^3\right)_{\sigma^3_3} \\ \hline 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \end{array} $$
$\sigma^3_1=\mathtt{0xAA}$ $\sigma^3_3=\mathtt{0xF0}$ $\sigma^3_2=\mathtt{0xCC}$
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