Minimize the following Boolean expression using Boolean algebra: [tex]$\overline{A B+A \bar{B} \bar{C}}+\bar{A} B+B C \bar{D}+\overline{B \bar{C}} \bar{D}$[/tex]
Answer (2)
Apply DeMorgan's Law to simplify the expression.
Expand and rearrange terms.
Use Boolean algebra rules to further simplify the expression.
The minimized Boolean expression is A ˉ + B ˉ C + C D ˉ + B ˉ D ˉ .
Explanation
Problem Analysis We are given the Boolean expression A B + A B ˉ C ˉ + A ˉ B + BC D ˉ + B C ˉ D ˉ and we want to minimize it using Boolean algebra.
Applying DeMorgan's Law First, we apply DeMorgan's Law to the first term: A B + A B ˉ C ˉ = A B ⋅ A B ˉ C ˉ = ( A ˉ + B ˉ ) ( A ˉ + B + C ) .
Expanding the Expression Next, we expand the first term: ( A ˉ + B ˉ ) ( A ˉ + B + C ) = A ˉ A ˉ + A ˉ B + A ˉ C + B ˉ A ˉ + B ˉ B + B ˉ C = A ˉ + A ˉ B + A ˉ C + A ˉ B ˉ + 0 + B ˉ C = A ˉ + A ˉ C + A ˉ B ˉ + B ˉ C = A ˉ ( 1 + C + B ˉ ) + B ˉ C = A ˉ + B ˉ C
Applying DeMorgan's Law Again Now, we apply DeMorgan's Law to the last term: B C ˉ = B ˉ + C . Thus, B C ˉ D ˉ = ( B ˉ + C ) D ˉ = B ˉ D ˉ + C D ˉ
Substituting Back Substitute the simplified terms back into the original expression: A ˉ + B ˉ C + A ˉ B + BC D ˉ + B ˉ D ˉ + C D ˉ
Rearranging Terms Rearrange the terms: A ˉ + A ˉ B + B ˉ C + BC D ˉ + C D ˉ + B ˉ D ˉ
Simplifying Simplify: A ˉ ( 1 + B ) + B ˉ C + C D ˉ ( B + 1 ) + B ˉ D ˉ = A ˉ + B ˉ C + C D ˉ + B ˉ D ˉ
Further Simplification Simplify further: A ˉ + B ˉ ( C + D ˉ ) + C D ˉ = A ˉ + B ˉ C + B ˉ D ˉ + C D ˉ
Final Simplified Expression The final simplified expression is: A ˉ + C D ˉ + B ˉ ( C + D ˉ ) or A ˉ + B ˉ C + C D ˉ + B ˉ D ˉ
Conclusion The minimized Boolean expression is A ˉ + B ˉ C + C D ˉ + B ˉ D ˉ .
Examples
Boolean algebra is used extensively in digital circuit design to simplify and optimize logic circuits. For example, minimizing a Boolean expression can reduce the number of logic gates required to implement a circuit, leading to lower cost, smaller size, and improved performance. This is crucial in designing efficient computer processors, memory units, and other digital systems.
The minimized Boolean expression is A ˉ + B ˉ C + C D ˉ + B ˉ D ˉ . This result is achieved by applying DeMorgan's Law and simplifying through Boolean algebra rules. The final expression minimizes the original complex form while retaining its logical equivalence.
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