12,080
edits
Changes
MyWikiBiz, Author Your Legacy — Wednesday May 22, 2024
Jump to navigationJump to searchDirectory talk:Jon Awbrey/Papers/Differential Logic : Introduction (view source)
Revision as of 15:28, 2 July 2009
, 15:28, 2 July 2009+ tables
| Not just one of the k arguments is true.
| Not just one of the k arguments is true.
|}
|}
===Tables===
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
|+ <math>\text{Table 1.}~~\text{Syntax and Semantics of a Calculus for Propositional Logic}</math>
|- style="background:#f0f0ff"
| <math>\text{Expression}\!</math>
| <math>\text{Interpretation}\!</math>
| <math>\text{Other Notations}\!</math>
|-
| <math>~</math>
| <math>\operatorname{true}</math>
| <math>1\!</math>
|-
| <math>\texttt{(~)}</math>
| <math>\operatorname{false}</math>
| <math>0\!</math>
|-
| <math>a\!</math>
| <math>a\!</math>
| <math>a\!</math>
|-
| <math>\texttt{(} a \texttt{)}</math>
| <math>\operatorname{not}~ a</math>
|
<math>\begin{matrix}
\lnot a & \bar{a} & \tilde{a} & a^\prime
\end{matrix}</math>
|-
| <math>a ~ b ~ c</math>
| <math>a ~\operatorname{and}~ b ~\operatorname{and}~ c</math>
| <math>a \land b \land c</math>
|-
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
| <math>a ~\operatorname{or}~ b ~\operatorname{or}~ c</math>
| <math>a \lor b \lor c</math>
|-
| <math>\texttt{(} a \texttt{(} b \texttt{))}</math>
|
<math>\begin{matrix}
a ~\operatorname{implies}~ b
\\
\operatorname{if}~ a ~\operatorname{then}~ b
\end{matrix}</math>
| <math>a \Rightarrow b</math>
|-
| <math>\texttt{(} a, b \texttt{)}</math>
|
<math>\begin{matrix}
a ~\operatorname{not~equal~to}~ b
\\
a ~\operatorname{exclusive~or}~ b
\end{matrix}</math>
|
<math>\begin{matrix}
a \neq b
\\
a + b
\end{matrix}</math>
|-
| <math>\texttt{((} a, b \texttt{))}</math>
|
<math>\begin{matrix}
a ~\operatorname{is~equal~to}~ b
\\
a ~\operatorname{if~and~only~if}~ b
\end{matrix}</math>
|
<math>\begin{matrix}
a = b
\\
a \Leftrightarrow b
\end{matrix}</math>
|-
| <math>\texttt{(} a, b, c \texttt{)}</math>
|
<math>\begin{matrix}
\operatorname{just~one~of}
\\
a, b, c
\\
\operatorname{is~false}.
\end{matrix}</math>
|
<math>\begin{matrix}
& \bar{a} ~ b ~ c
\\
\lor & a ~ \bar{b} ~ c
\\
\lor & a ~ b ~ \bar{c}
\end{matrix}</math>
|-
| <math>\texttt{((} a \texttt{),(} b \texttt{),(} c \texttt{))}</math>
|
<math>\begin{matrix}
\operatorname{just~one~of}
\\
a, b, c
\\
\operatorname{is~true}.
\\[6pt]
\operatorname{partition~all}
\\
\operatorname{into}~ a, b, c.
\end{matrix}</math>
|
<math>\begin{matrix}
& a ~ \bar{b} ~ \bar{c}
\\
\lor & \bar{a} ~ b ~ \bar{c}
\\
\lor & \bar{a} ~ \bar{b} ~ c
\end{matrix}</math>
|-
|
<math>\begin{matrix}
\texttt{((} a, b \texttt{)}, c \texttt{)}
\\[6pt]
\texttt{(} a, \texttt{(} b, c \texttt{))}
\end{matrix}</math>
|
<math>\begin{matrix}
\operatorname{oddly~many~of}
\\
a, b, c
\\
\operatorname{are~true}.
\end{matrix}</math>
|
<p><math>a + b + c\!</math></p>
<br>
<p><math>\begin{matrix}
& a ~ b ~ c
\\
\lor & a ~ \bar{b} ~ \bar{c}
\\
\lor & \bar{a} ~ b ~ \bar{c}
\\
\lor & \bar{a} ~ \bar{b} ~ c
\end{matrix}</math></p>
|-
| <math>\texttt{(} x, \texttt{(} a \texttt{),(} b \texttt{),(} c \texttt{))}</math>
|
<math>\begin{matrix}
\operatorname{partition}~ x
\\
\operatorname{into}~ a, b, c.
\\[6pt]
\operatorname{genus}~ x ~\operatorname{comprises}
\\
\operatorname{species}~ a, b, c.
\end{matrix}</math>
|
<math>\begin{matrix}
& \bar{x} ~ \bar{a} ~ \bar{b} ~ \bar{c}
\\
\lor & x ~ a ~ \bar{b} ~ \bar{c}
\\
\lor & x ~ \bar{a} ~ b ~ \bar{c}
\\
\lor & x ~ \bar{a} ~ \bar{b} ~ c
\end{matrix}</math>
|}
<br>
