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| | 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> |