... # Useful NAND building blocks **What is the Boolean equation for an inverter using only NAND logic?** The Boolean equation for an inverter using only NAND logic is $(A\cdot A)'$. **What is the Boolean equation for an AND gate using only NAND logic?** The Boolean equation for an AND gate using only NAND logic is $((A\cdot B)'(A\cdot B)')'$. **What is the Boolean equation for an OR gate using only NAND logic?** The Boolean equation for an OR gate using only NAND logic is $((A\cdot A)'(B\cdot B)')'$. **What is the Boolean equation for a NOR gate using only NAND logic?** The Boolean equation for a NOR gate using only NAND logic is $((A+B)(A+B))'$. > [!summary] > NAND Inverter: $(A\cdot A)'$. > NAND AND: $((A\cdot B)'(A\cdot B)')$. > NAND OR: $((A\cdot A)'(B\cdot B)')$. > NAND NOR: $((A+B)(A+B))'$. # Useful NOR building blocks **What is the Boolean equation for an inverter using only NOR logic?** The Boolean equation for an inverter using only NOR logic is *$(A+A)'$.* **What is the Boolean equation for an AND gate using only NOR logic?** The Boolean equation for an AND gate using only NOR logic is *$((A+A)'+(B+B)')'$.* **What is the Boolean equation for an OR gate using only NOR logic?** The Boolean equation for an OR gate using only NOR logic is *$((A+B)'+(A+B)')'$.* **What is the Boolean equation for a NAND gate using only NOR logic?** The Boolean equation for a NAND gate using only NOR logic is *$((A\cdot B)+(A\cdot B))'$.* > [!summary] > NOR Inverter: $(A+A)'$. > NOR AND: $((A+A)'+(B+B)')'$. > NOR OR: $((A+B)'+(A+B)')'$. > NOR NAND: $((A\cdot B)+(A\cdot B))'$. # NAND / NOR Implementation **What type of Boolean equation should you convert to NAND logic?** The type of Boolean equation you should convert to NAND logic is *Sum of Product (SOP).* **What type of Boolean equation should you convert to NOR logic?** The type of Boolean equation you should convert to NOR logic is *Product of Sum (POS).* > [!summary] > SOP equations should convert to NAND. > POS equations should convert to NOR. # Back to basics: forming Boolean function from truth table **In the equation $f(x_1,x_2,x_3)=\sum\textrm{m}(0,1,5)$, what do the numbers within $m(\dots)$ represent?** In the equation $f(x_1,x_2,x_3)=\sum\textrm{m}(0,1,5)$, the numbers within $\textrm{m}(\dots)$ represent *the row numbers of the truth table where the minterm $x_1\cdot x_2\cdot x_3=1$.* ... # K-Map **What are the coordinates of each cell in a 4x4 Karnaugh map?** The coordinates of each cell in a 4x4 Karnaugh map are: | $X_3X_4$\\$X_1X_2$ | 00 | 01 | 11 | 10 | | ------------------ | --- | --- | --- | --- | | 00 | 0 | 4 | 12 | 8 | | 01 | 1 | 5 | 13 | 9 | | 11 | 3 | 7 | 15 | 11 | | 10 | 2 | 6 | 14 | 10 | **What do the coordinates of each cell in a Karnaugh map correspond to?** The coordinates of each cell in a Karnaugh map correspond to *the row number of a truth table.* **Can you group the corners of a Karnaugh map together horizontally / vertically?** *Yes*, you can group the corners of a Karnaugh map together horizontally / vertically.