Using NAND gates

NOT

Figure 12.10: Realizing a NOT gate using a NAND gate

OR The following statements are called DeMorgan's Theorems and can be easily verified and extended for more than two variables.

$$\overline{A+B}=\overline{A}.\overline{B}$$ (12.1)

$$\overline{A.B}=\overline{A}+\overline{B}$$ (12.2)

$$(X+Y)(X+Z)=X.X+X.Z+Y.X+Y.Z$$ (12.3)

$$=X+X(Z+Y)+Y.Z$$ (12.4)

$$X+X.F=X(1+F)=X$$ (12.5)

$$(X+Y)(X+Z)=X+YZ$$ (12.6)

Now it is easy to see that $A+\overline{A}B=A+B$, which can be checked from the truth table easily. The resulting realization of OR gate is shown in 12.11

Figure 12.11: Realization of OR gate by NAND gates

AND gate

Figure 12.12: Realization of AND gate by NAND gates

X-OR gate

$$C=\overline{A}B+A\overline{B}$$ (12.7)

Clearly, this can be implemented using AND, NOT and OR gates, and hence can be implemented using universal gates.

Figure 12.13: X-OR gate

X-NOR gate

$$C=\overline{A}\overline{B}+AB$$ (12.8)

Again, this can be implemented using AND, NOT and OR gates, and hence can be implemented using universal gates, i.e., NAND or NOR gates.

Figure 12.14: X-NOR gate