Minimize the following Boolean expression: [tex]$\overline{A B+A \bar{B} \bar{C}+\bar{A} B+B \bar{C} \bar{D}+\overline{B C} \bar{D}}$[/tex]
Answer (2)
The minimized Boolean expression is D ⋅ B ˉ ⋅ ( C + A ˉ ) . This was achieved by applying DeMorgan's Law, expanding, and simplifying the terms step by step. The final expression reduces the complexity of the original expression while maintaining the same logic.
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Apply DeMorgan's Law to transform the expression.
Apply DeMorgan's Law again to further expand the terms.
Substitute the simplified terms back into the expression.
Simplify the expression to obtain the minimized form: D B ˉ ( C + A ˉ )
Explanation
Problem Analysis We are given the Boolean expression A B + A B ˉ C ˉ + A ˉ B + B C ˉ D ˉ + BC D ˉ and our objective is to minimize it.
Applying DeMorgan's Law First, we apply DeMorgan's Law to the entire expression. DeMorgan's Law states that X + Y = X ˉ ⋅ Y ˉ . Applying this, we get:
A B + A B ˉ C ˉ + A ˉ B + B C ˉ D ˉ + BC D ˉ = A B ⋅ A B ˉ C ˉ ⋅ A ˉ B ⋅ B C ˉ D ˉ ⋅ BC D ˉ
Applying DeMorgan's Law Again Next, we apply DeMorgan's Law again to each term. We use the fact that X Y = X ˉ + Y ˉ .
A B = A ˉ + B ˉ
A B ˉ C ˉ = A ˉ + B + C
A ˉ B = A + B ˉ
B C ˉ D ˉ = B ˉ + C + D
BC D ˉ = BC + D
Substituting Back Now, we substitute these back into the expression:
( A ˉ + B ˉ ) ( A ˉ + B + C ) ( A + B ˉ ) ( B ˉ + C + D ) ( BC + D )
Simplified Expression After simplification using a tool, the minimized expression is:
D ⋅ B ⋅ ( C + A )
Final Answer Therefore, the minimized Boolean expression is D B ˉ ( C + A ˉ )
Examples
Boolean algebra is used extensively in digital circuit design. Minimizing Boolean expressions allows engineers to simplify the logic gates required to implement a particular function, which reduces the cost and complexity of the circuit. For example, in designing a circuit for a washing machine, Boolean algebra can be used to control the water level, temperature, and spin cycle based on user inputs and sensor readings. By minimizing the Boolean expressions, the circuit can be made more efficient and reliable.
