The simplified form of a Boolean equation $(A\overline{B} + A\overline{B}C + AC)(\overline{A}\overline{C} + \overline{B})$ is :
Simplifying the Boolean Equation
We need to simplify the Boolean expression: $(A\overline{B} + A\overline{B}C + AC)(\overline{A}\overline{C} + \overline{B})$
Step 1: Simplify the first parenthesis
- Original term: $A\overline{B} + A\overline{B}C + AC$
- Apply the absorption law ($X + XY = X$) to the first two terms: $A\overline{B} + A\overline{B}C = A\overline{B}$.
- The first parenthesis simplifies to: $A\overline{B} + AC$.
Step 2: Expand the expression
Now, substitute the simplified first parenthesis back into the original equation:
$(A\overline{B} + AC)(\overline{A}\overline{C} + \overline{B})$Expand using the distributive property (FOIL):
$ (A\overline{B} \cdot \overline{A}\overline{C}) + (A\overline{B} \cdot \overline{B}) + (AC \cdot \overline{A}\overline{C}) + (AC \cdot \overline{B}) $Step 3: Apply Boolean Laws
Simplify each term using Boolean algebra laws:
- $A\overline{B} \cdot \overline{A}\overline{C} = (A \cdot \overline{A}) \overline{B} \overline{C} = 0 \cdot \overline{B} \overline{C} = 0$ (using $X \cdot \overline{X} = 0$)
- $A\overline{B} \cdot \overline{B} = A (\overline{B} \cdot \overline{B}) = A\overline{B}$ (using $X \cdot X = X$)
- $AC \cdot \overline{A}\overline{C} = (A \cdot \overline{A}) (C \cdot \overline{C}) = 0 \cdot 0 = 0$
- $AC \cdot \overline{B} = A\overline{B}C
Step 4: Combine simplified terms
Add the simplified terms together:
$ 0 + A\overline{B} + 0 + A\overline{B}C $This simplifies to:
$ A\overline{B} + A\overline{B}C $Step 5: Final Simplification
Apply the absorption law ($X + XY = X$) again, where $X = A\overline{B}$:
$ A\overline{B} + (A\overline{B})C = A\overline{B} $Conclusion
The simplified form of the Boolean equation is $A\overline{B}$.
Important Questions from Minimization of Boolean Expression
Consider the following four variable Boolean function in sum-of-product form
$F(b_3, b_2, b_1, b_0) = \sum(0, 2, 4, 8, 10, 11, 12)$.
where the value of the function is computed by considering $b_3b_2b_1b_0$ as a 4-bit binary number, where $b_3$ denotes the most significant bit and $b_0$ denotes the least significant bit. Note that there are no don't care terms. Which ONE of the following options is the CORRECT minimized Boolean expression for F?
- Simplified Boolean equation for the following truth table is :
x y z F 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 1 1 0 0 1 1 0 1 0 1 1 0 1 1 1 1 0 Simplify the following Boolean expression.
E(E + F) + DE + D(E + F)
What is the value of \( \bar{F}\)?
\(F = AB + \bar{C}\bar{D} + \bar{B}D\)
Which statement(s) is/are correct regarding the Boolean algebra?
I. It facilitate the analysis and design of digital circuits.
II. Expresses in algebraic form the input-output relationship of logic diagram.
