Find equivalent circuits for the following with a lesser number of logic gates
Answer
Simplified Expression: out=ABC\text{out} = ABCout=ABC
Simplified Circuit:
A single three-input AND gate with inputs A, B, and C.
Explanation
The given logic expression is: out=A(BC+BC+BC)+B(AC+CA)\text{out} = A(BC + BC + BC) + B(AC + CA)out=A(BC+BC+BC)+B(AC+CA)
First, combine like terms inside the parentheses: BC+BC+BC=BCAC+CA=ACBC + BC + BC = BC AC + CA = AC BC+BC+BC=BCAC+CA=AC
So the expression becomes: out=A(BC)+B(AC)\text{out} = A(BC) + B(AC) out=A(BC)+B(AC)
Next, distribute the terms: out=ABC+ABC\text{out} = ABC + ABC out=ABC+ABC
Since both terms are identical (ABC), we can eliminate redundancy: out=ABC\text{out} = ABC out=ABC
This is the simplest form of the given logic function. The Boolean laws applied include Idempotent Law (X + X = X) and Associative Law. The original circuit used multiple AND and OR gates because the expression was unsimplified. By reducing the expression, we minimize the hardware needed.
The simplified circuit requires only one AND gate with three inputs: A, B, and C. This design is more efficient because it reduces the gate count, power consumption, and complexity of the circuit. Fewer components mean lower cost and less space on the chip, which is crucial in digital design.
Using Boolean algebra to simplify logic expressions is an essential practice in digital electronics. Simplification not only optimizes the physical layout but also improves speed since signals pass through fewer gates. Additionally, minimizing logic gates can enhance reliability because fewer components reduce the chance of failure.
In conclusion, the original expression was unnecessarily complex due to repeated terms. After simplification, the logic function turned into a single product term, ABC, implemented using one three-input AND gate. This is the most efficient solution for the given problem.