Line 3,063: |
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| By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions in a number of different languages for zeroth order logic. | | By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions in a number of different languages for zeroth order logic. |
| | | |
− | {| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" | + | {| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" |
− | |+ '''Table 1. Propositional Forms on Two Variables''' | + | |+ '''Table 2. Propositional Forms on Two Variables''' |
− | |- style="background:ghostwhite" | + | |- style="background:ghostwhite; height:36px" |
| | <math>\mathcal{L}_1</math> | | | <math>\mathcal{L}_1</math> |
| | <math>\mathcal{L}_2</math> | | | <math>\mathcal{L}_2</math> |
Line 3,072: |
Line 3,072: |
| | <math>\mathcal{L}_5</math> | | | <math>\mathcal{L}_5</math> |
| | <math>\mathcal{L}_6</math> | | | <math>\mathcal{L}_6</math> |
− | |- style="background:ghostwhite" | + | |- style="background:ghostwhite; height:48px" |
| | | | | |
− | | align="right" | <math>x\!</math> : | + | | |
− | | 1 1 0 0 | + | {| align="right" style="background:ghostwhite; text-align:right" |
− | | | + | |- |
− | | | + | | <math>x\!</math> : |
− | | | + | |- |
− | |- style="background:ghostwhite" | + | | <math>y\!</math> : |
− | | | + | |} |
− | | align="right" | <math>y\!</math> : | + | | |
| + | {| align="center" style="background:ghostwhite" |
| + | |- |
| + | | 1 1 0 0 |
| + | |- |
| | 1 0 1 0 | | | 1 0 1 0 |
| + | |} |
| | | | | |
| | | | | |
| | | | | |
| |- | | |- |
− | | <math>f_{0}\!</math> | + | | |
− | | <math>f_{0000}\!</math> | + | {| align="center" |
− | | 0 0 0 0 | + | |- |
− | | <math>(~)\!</math> | + | | height="36px" | <p><math>f_{0}\!</math></p> |
− | | <math>\operatorname{false}</math> | + | |- |
− | | <math>0\!</math> | + | | height="36px" | <p><math>f_{1}\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>f_{2}\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>f_{3}\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>f_{4}\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>f_{5}\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>f_{6}\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>f_{7}\!</math></p> |
| + | |} |
| + | | |
| + | {| align="center" |
| + | |- |
| + | | height="36px" | <p><math>f_{0000}\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>f_{0001}\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>f_{0010}\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>f_{0011}\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>f_{0100}\!</math></p> |
| |- | | |- |
− | | <math>f_{1}\!</math> | + | | height="36px" | <p><math>f_{0101}\!</math></p> |
− | | <math>f_{0001}\!</math>
| |
− | | 0 0 0 1
| |
− | | <math>(x)(y)\!</math>
| |
− | | <math>\operatorname{neither}\ x\ \operatorname{nor}\ y</math>
| |
− | | <math>\lnot x \land \lnot y\!</math>
| |
| |- | | |- |
− | | <math>f_{2}\!</math> | + | | height="36px" | <p><math>f_{0110}\!</math></p> |
− | | <math>f_{0010}\!</math>
| |
− | | 0 0 1 0
| |
− | | <math>(x)\ y\!</math>
| |
− | | <math>y\ \operatorname{without}\ x</math>
| |
− | | <math>\lnot x \land y\!</math>
| |
| |- | | |- |
− | | <math>f_{3}\!</math> | + | | height="36px" | <p><math>f_{0111}\!</math></p> |
− | | <math>f_{0011}\!</math>
| + | |} |
− | | 0 0 1 1 | + | | |
− | | <math>(x)\!</math> | + | {| align="center" |
− | | <math>\operatorname{not}\ x</math>
| |
− | | <math>\lnot x\!</math> | |
| |- | | |- |
− | | <math>f_{4}\!</math> | + | | height="36px" | 0 0 0 0 |
− | | <math>f_{0100}\!</math>
| |
− | | 0 1 0 0 | |
− | | <math>x\ (y)\!</math>
| |
− | | <math>x\ \operatorname{without}\ y</math>
| |
− | | <math>x \land \lnot y\!</math>
| |
| |- | | |- |
− | | <math>f_{5}\!</math> | + | | height="36px" | 0 0 0 1 |
− | | <math>f_{0101}\!</math>
| |
− | | 0 1 0 1 | |
− | | <math>(y)\!</math>
| |
− | | <math>\operatorname{not}\ y</math>
| |
− | | <math>\lnot y\!</math>
| |
| |- | | |- |
− | | <math>f_{6}\!</math> | + | | height="36px" | 0 0 1 0 |
− | | <math>f_{0110}\!</math>
| |
− | | 0 1 1 0 | |
− | | <math>(x,\ y)\!</math>
| |
− | | <math>x\ \operatorname{not~equal~to}\ y</math>
| |
− | | <math>x \ne y\!</math>
| |
| |- | | |- |
− | | <math>f_{7}\!</math> | + | | height="36px" | 0 0 1 1 |
− | | <math>f_{0111}\!</math>
| |
− | | 0 1 1 1 | |
− | | <math>(x\ y)\!</math>
| |
− | | <math>\operatorname{not~both}\ x\ \operatorname{and}\ y</math>
| |
− | | <math>\lnot x \lor \lnot y\!</math>
| |
| |- | | |- |
− | | <math>f_{8}\!</math> | + | | height="36px" | 0 1 0 0 |
− | | <math>f_{1000}\!</math>
| |
− | | 1 0 0 0 | |
− | | <math>x\ y\!</math>
| |
− | | <math>x\ \operatorname{and}\ y</math>
| |
− | | <math>x \land y\!</math>
| |
| |- | | |- |
− | | <math>f_{9}\!</math> | + | | height="36px" | 0 1 0 1 |
− | | <math>f_{1001}\!</math>
| |
− | | 1 0 0 1 | |
− | | <math>((x,\ y))\!</math>
| |
− | | <math>x\ \operatorname{equal~to}\ y</math>
| |
− | | <math>x = y\!</math>
| |
| |- | | |- |
− | | <math>f_{10}\!</math> | + | | height="36px" | 0 1 1 0 |
− | | <math>f_{1010}\!</math> | + | |- |
− | | 1 0 1 0 | + | | height="36px" | 0 1 1 1 |
− | | <math>y\!</math> | + | |} |
− | | <math>y\!</math> | + | | |
− | | <math>y\!</math> | + | {| align="center" |
| + | |- |
| + | | height="36px" | <p><math>(~)\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>(x)(y)\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>(x)\ y\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>(x)\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>x\ (y)\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>(y)\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>(x,\ y)\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>(x\ y)\!</math></p> |
| + | |} |
| + | | |
| + | {| align="center" |
| + | |- |
| + | | height="36px" | <p><math>\operatorname{false}</math></p> |
| + | |- |
| + | | height="36px" | <p><math>\operatorname{neither}\ x\ \operatorname{nor}\ y</math></p> |
| + | |- |
| + | | height="36px" | <p><math>y\ \operatorname{without}\ x</math></p> |
| + | |- |
| + | | height="36px" | <p><math>\operatorname{not}\ x</math></p> |
| + | |- |
| + | | height="36px" | <p><math>x\ \operatorname{without}\ y</math></p> |
| + | |- |
| + | | height="36px" | <p><math>\operatorname{not}\ y</math></p> |
| + | |- |
| + | | height="36px" | <p><math>x\ \operatorname{not~equal~to}\ y</math></p> |
| + | |- |
| + | | height="36px" | <p><math>\operatorname{not~both}\ x\ \operatorname{and}\ y</math></p> |
| + | |} |
| + | | |
| + | {| align="center" |
| + | |- |
| + | | height="36px" | <p><math>0\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>\lnot x \land \lnot y</math></p> |
| + | |- |
| + | | height="36px" | <p><math>\lnot x \land y</math></p> |
| + | |- |
| + | | height="36px" | <p><math>\lnot x</math></p> |
| + | |- |
| + | | height="36px" | <p><math>x \land \lnot y</math></p> |
| + | |- |
| + | | height="36px" | <p><math>\lnot y</math></p> |
| + | |- |
| + | | height="36px" | <p><math>x \ne y</math></p> |
| + | |- |
| + | | height="36px" | <p><math>\lnot x \lor \lnot y</math></p> |
| + | |} |
| + | |- |
| + | | |
| + | {| align="center" |
| + | |- |
| + | | height="36px" | <p><math>f_{8}\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>f_{9}\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>f_{10}\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>f_{11}\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>f_{12}\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>f_{13}\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>f_{14}\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>f_{15}\!</math></p> |
| + | |} |
| + | | |
| + | {| align="center" |
| + | |- |
| + | | height="36px" | <p><math>f_{1000}\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>f_{1001}\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>f_{1010}\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>f_{1011}\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>f_{1100}\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>f_{1101}\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>f_{1110}\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>f_{1111}\!</math></p> |
| + | |} |
| + | | |
| + | {| align="center" |
| + | |- |
| + | | height="36px" | 1 0 0 0 |
| + | |- |
| + | | height="36px" | 1 0 0 1 |
| + | |- |
| + | | height="36px" | 1 0 1 0 |
| + | |- |
| + | | height="36px" | 1 0 1 1 |
| + | |- |
| + | | height="36px" | 1 1 0 0 |
| + | |- |
| + | | height="36px" | 1 1 0 1 |
| + | |- |
| + | | height="36px" | 1 1 1 0 |
| + | |- |
| + | | height="36px" | 1 1 1 1 |
| + | |} |
| + | | |
| + | {| align="center" |
| + | |- |
| + | | height="36px" | <p><math>x\ y\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>((x,\ y))\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>y\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>(x\ (y))\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>x\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>((x)\ y)\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>((x)(y))\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>((~))\!</math></p> |
| + | |} |
| + | | |
| + | {| align="center" |
| + | |- |
| + | | height="36px" | <p><math>x\ \operatorname{and}\ y</math></p> |
| + | |- |
| + | | height="36px" | <p><math>x\ \operatorname{equal~to}\ y</math></p> |
| + | |- |
| + | | height="36px" | <p><math>y\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>\operatorname{not}\ x\ \operatorname{without}\ y</math></p> |
| + | |- |
| + | | height="36px" | <p><math>x\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>\operatorname{not}\ y\ \operatorname{without}\ x</math></p> |
| + | |- |
| + | | height="36px" | <p><math>x\ \operatorname{or}\ y</math></p> |
| + | |- |
| + | | height="36px" | <p><math>\operatorname{true}</math></p> |
| + | |} |
| + | | |
| + | {| align="center" |
| + | |- |
| + | | height="36px" | <p><math>x \land y</math></p> |
| + | |- |
| + | | height="36px" | <p><math>x = y\!</math></p> |
| + | |- |
| + | | height="36px" | <p><math>y\!</math></p> |
| |- | | |- |
− | | <math>f_{11}\!</math> | + | | height="36px" | <p><math>x \Rightarrow y</math></p> |
− | | <math>f_{1011}\!</math> | |
− | | 1 0 1 1
| |
− | | <math>(x\ (y))\!</math>
| |
− | | <math>\operatorname{not}\ x\ \operatorname{without}\ y</math>
| |
− | | <math>x \Rightarrow y\!</math>
| |
| |- | | |- |
− | | <math>f_{12}\!</math> | + | | height="36px" | <p><math>x\!</math></p> |
− | | <math>f_{1100}\!</math> | |
− | | 1 1 0 0
| |
− | | <math>x\!</math>
| |
− | | <math>x\!</math>
| |
− | | <math>x\!</math>
| |
| |- | | |- |
− | | <math>f_{13}\!</math> | + | | height="36px" | <p><math>x \Leftarrow y</math></p> |
− | | <math>f_{1101}\!</math> | |
− | | 1 1 0 1
| |
− | | <math>((x)\ y)\!</math>
| |
− | | <math>\operatorname{not}\ y\ \operatorname{without}\ x</math>
| |
− | | <math>x \Leftarrow y\!</math>
| |
| |- | | |- |
− | | <math>f_{14}\!</math> | + | | height="36px" | <p><math>x \lor y</math></p> |
− | | <math>f_{1110}\!</math> | |
− | | 1 1 1 0
| |
− | | <math>((x)(y))\!</math>
| |
− | | <math>x\ \operatorname{or}\ y</math>
| |
− | | <math>x \lor y\!</math>
| |
| |- | | |- |
− | | <math>f_{15}\!</math> | + | | height="36px" | <p><math>1\!</math></p> |
− | | <math>f_{1111}\!</math> | + | |} |
− | | 1 1 1 1
| |
− | | <math>((~))\!</math>
| |
− | | <math>\operatorname{true}</math>
| |
− | | <math>1\!</math> | |
| |} | | |} |
| <br> | | <br> |