Boolean Algebra and Simplification Techniques
203
(A+B)
A
B
Figure 6.4 Example 6.6.
Example 6.6
Starting with the Boolean expression for a two-input OR gate, apply Boolean laws and theorems to
modify it in such a way as to facilitate the implementation of a two-input OR gate by using two-input
NAND gates only.
Solution
• A two-input OR gate is represented by the Boolean equation Y = A + BB,
where A and B are the input logic variables and Y is the output.
• NowA + BB = A + BB
Involution law
= AABB
DeMorgan’s theorem
= AAAAAABBBBB Idempotent law
• Figure 6.4 shows the NAND gate implementation of a two-input OR gate.
Example 6.7
Apply suitable Boolean laws and theorems to modify the expression for a two-input EX-OR gate in
such a way as to implement a two-input EX-OR gate by using the minimum number of two-input NAND
gates only.
Solution
• A two-input EX-OR gate is represented by the Boolean expression Y = AAB + AAB.
• NowAAB + AAB = AAB + AAB
Involution law
= AABBAAB
DeMorgan’s law
= BBBA + BBBBBAAAA + BBB
= BBAABBBBAAAABB
B632
• Equation (6.32) is in a form that can be implemented with NAND gates only.
• Figure 6.5 shows the logic diagram.