Digital Electronics: Principles, Devices and Applications

Boolean Algebra and Simplification Techniques 203A(A+B)BFigure 6.4 Example 6.6.Example 6.6Starting with the Boolean expression for a two-input OR gate, apply Boolean laws and theorems tomodify it in such a way as to facilitate the implementation of a two-input OR gate by using two-inputNAND gates only.Solution• A two-input OR gate is represented by the Boolean equation Y = A + B,where A and B are the input logic variables and Y is the output.• NowA + B = A + B Involution law= AB DeMorgan’s theorem= AABB Idempotent law• Figure 6.4 shows the NAND gate implementation of a two-input OR gate.Example 6.7Apply suitable Boolean laws and theorems to modify the expression for a two-input EX-OR gate insuch a way as to implement a two-input EX-OR gate by using the minimum number of two-input NANDgates only.Solution• A two-input EX-OR gate is represented by the Boolean expression Y = AB + AB.• NowAB + AB = AB + ABInvolution law= ABAB DeMorgan’s law= BA + BAA + B= BABAAB 632• Equation (6.32) is in a form that can be implemented with NAND gates only.• Figure 6.5 shows the logic diagram.