| Line 181: |
Line 181: |
| | The 16 boolean functions on two variables, <math>F^{(2)} : \mathbb{B}^2 \to \mathbb{B},</math> are shown in the following Table. | | The 16 boolean functions on two variables, <math>F^{(2)} : \mathbb{B}^2 \to \mathbb{B},</math> are shown in the following Table. |
| | | | |
| − | <br>
| + | {| align="center" cellpadding="0" cellspacing="0" style="text-align:center" |
| − | | + | |+ style="height:25px; font-size:large" | <math>\text{Table 18. Boolean Functions on Two Variables}</math> |
| − | {| align="center" border="1" cellpadding="4" cellspacing="0" style="text-align:center; width:60%" | |
| − | |+ style="height:30px" | <math>\text{Table 18. Boolean Functions on Two Variables}</math> | |
| − | |- style="height:40px; background:ghostwhite"
| |
| − | | width="14%" | <math>F</math>
| |
| − | | width="14%" | <math>F</math>
| |
| − | | colspan="4" | <math>F(x, y)</math>
| |
| − | | width="24%" | <math>F</math>
| |
| − | |- style="height:40px; background:ghostwhite"
| |
| − | | width="14%" |
| |
| − | | width="14%" |
| |
| − | | width="12%" | <math>F(1, 1)</math>
| |
| − | | width="12%" | <math>F(1, 0)</math>
| |
| − | | width="12%" | <math>F(0, 1)</math>
| |
| − | | width="12%" | <math>F(0, 0)</math>
| |
| − | | width="24%" |
| |
| | |- | | |- |
| − | | <math>F_{0}^{(2)}</math> | + | | [[File:Boolean Functions on Two Variables • Truth Table.png|600px]] |
| − | | <math>F_{0000}^{(2)}</math> | |
| − | | <math>0</math>
| |
| − | | <math>0</math>
| |
| − | | <math>0</math>
| |
| − | | <math>0</math>
| |
| − | | <math>\texttt{( )}</math>
| |
| − | |-
| |
| − | | <math>F_{1}^{(2)}</math>
| |
| − | | <math>F_{0001}^{(2)}</math>
| |
| − | | <math>0</math>
| |
| − | | <math>0</math>
| |
| − | | <math>0</math>
| |
| − | | <math>1</math>
| |
| − | | <math>\texttt{(} x \texttt{)(} y \texttt{)}</math>
| |
| − | |-
| |
| − | | <math>F_{2}^{(2)}</math>
| |
| − | | <math>F_{0010}^{(2)}</math>
| |
| − | | <math>0</math>
| |
| − | | <math>0</math>
| |
| − | | <math>1</math>
| |
| − | | <math>0</math>
| |
| − | | <math>\texttt{(} x \texttt{)} y</math>
| |
| − | |-
| |
| − | | <math>F_{3}^{(2)}</math>
| |
| − | | <math>F_{0011}^{(2)}</math>
| |
| − | | <math>0</math>
| |
| − | | <math>0</math>
| |
| − | | <math>1</math>
| |
| − | | <math>1</math>
| |
| − | | <math>\texttt{(} x \texttt{)}</math>
| |
| − | |-
| |
| − | | <math>F_{4}^{(2)}</math>
| |
| − | | <math>F_{0100}^{(2)}</math>
| |
| − | | <math>0</math>
| |
| − | | <math>1</math>
| |
| − | | <math>0</math>
| |
| − | | <math>0</math>
| |
| − | | <math>x \texttt{(} y \texttt{)}</math>
| |
| − | |-
| |
| − | | <math>F_{5}^{(2)}</math>
| |
| − | | <math>F_{0101}^{(2)}</math>
| |
| − | | <math>0</math>
| |
| − | | <math>1</math>
| |
| − | | <math>0</math>
| |
| − | | <math>1</math>
| |
| − | | <math>\texttt{(} y \texttt{)}</math>
| |
| − | |-
| |
| − | | <math>F_{6}^{(2)}</math>
| |
| − | | <math>F_{0110}^{(2)}</math>
| |
| − | | <math>0</math>
| |
| − | | <math>1</math>
| |
| − | | <math>1</math>
| |
| − | | <math>0</math>
| |
| − | | <math>\texttt{(} x \texttt{,} y \texttt{)}</math>
| |
| − | |-
| |
| − | | <math>F_{7}^{(2)}</math>
| |
| − | | <math>F_{0111}^{(2)}</math>
| |
| − | | <math>0</math>
| |
| − | | <math>1</math>
| |
| − | | <math>1</math>
| |
| − | | <math>1</math>
| |
| − | | <math>\texttt{(} x y \texttt{)}</math>
| |
| − | |-
| |
| − | | <math>F_{8}^{(2)}</math>
| |
| − | | <math>F_{1000}^{(2)}</math>
| |
| − | | <math>1</math>
| |
| − | | <math>0</math>
| |
| − | | <math>0</math>
| |
| − | | <math>0</math>
| |
| − | | <math>x y</math>
| |
| − | |-
| |
| − | | <math>F_{9}^{(2)}</math>
| |
| − | | <math>F_{1001}^{(2)}</math>
| |
| − | | <math>1</math>
| |
| − | | <math>0</math>
| |
| − | | <math>0</math>
| |
| − | | <math>1</math>
| |
| − | | <math>\texttt{((} x \texttt{,} y \texttt{))}</math>
| |
| − | |-
| |
| − | | <math>F_{10}^{(2)}</math>
| |
| − | | <math>F_{1010}^{(2)}</math>
| |
| − | | <math>1</math>
| |
| − | | <math>0</math>
| |
| − | | <math>1</math>
| |
| − | | <math>0</math>
| |
| − | | <math>y</math>
| |
| − | |-
| |
| − | | <math>F_{11}^{(2)}</math>
| |
| − | | <math>F_{1011}^{(2)}</math>
| |
| − | | <math>1</math>
| |
| − | | <math>0</math>
| |
| − | | <math>1</math>
| |
| − | | <math>1</math>
| |
| − | | <math>\texttt{(} x \texttt{(} y \texttt{))}</math>
| |
| − | |-
| |
| − | | <math>F_{12}^{(2)}</math>
| |
| − | | <math>F_{1100}^{(2)}</math>
| |
| − | | <math>1</math>
| |
| − | | <math>1</math>
| |
| − | | <math>0</math>
| |
| − | | <math>0</math>
| |
| − | | <math>x</math>
| |
| − | |-
| |
| − | | <math>F_{13}^{(2)}</math>
| |
| − | | <math>F_{1101}^{(2)}</math>
| |
| − | | <math>1</math>
| |
| − | | <math>1</math>
| |
| − | | <math>0</math>
| |
| − | | <math>1</math>
| |
| − | | <math>\texttt{((} x \texttt{)} y \texttt{)}</math>
| |
| − | |-
| |
| − | | <math>F_{14}^{(2)}</math>
| |
| − | | <math>F_{1110}^{(2)}</math>
| |
| − | | <math>1</math>
| |
| − | | <math>1</math>
| |
| − | | <math>1</math>
| |
| − | | <math>0</math>
| |
| − | | <math>\texttt{((} x \texttt{)(} y \texttt{))}</math>
| |
| − | |-
| |
| − | | <math>F_{15}^{(2)}</math>
| |
| − | | <math>F_{1111}^{(2)}</math>
| |
| − | | <math>1</math>
| |
| − | | <math>1</math>
| |
| − | | <math>1</math>
| |
| − | | <math>1</math>
| |
| − | | <math>\texttt{(( ))}</math>
| |
| | |} | | |} |
| − |
| |
| − | <br>
| |
| | | | |
| | As before, all boolean functions on proper subsets of the current variables are subsumed in the Table at hand. In particular, we have the following inclusions. | | As before, all boolean functions on proper subsets of the current variables are subsumed in the Table at hand. In particular, we have the following inclusions. |