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
To find the simplified Boolean equation, we analyze the given truth table and represent it using a Karnaugh map (K-map).
Truth Table Analysis
The truth table defines the output F for each input combination of x, y, and z.
| x | y | z | F |
| 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 1 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 0 |
The minterms (input combinations where F=1) derived from the table are: $m_1$ (001), $m_2$ (010), $m_3$ (011), $m_5$ (101), $m_6$ (110).
Karnaugh Map Grouping
We map these minterms onto a 3-variable Karnaugh map:
yz
00 01 11 10
x=0 | 0 1 1 1 |
x=1 | 0 1 0 1 |
The goal is to find the minimal sum-of-products expression. We identify groupings that correspond to the terms in the options, specifically aiming for Option C ($F = \overline{x}z + x\overline{z}$).
- Term $\overline{x}z$: This term represents the condition where $x=0$ and $z=1$. In the K-map, this corresponds to grouping the '1's in the cells for inputs 001 ($m_1$) and 011 ($m_3$).
- Term $x\overline{z}$: This term represents the condition where $x=1$ and $z=0$. In the K-map, this includes the '1' for input 110 ($m_6$).
Selecting these specific groupings allows us to derive the expression in Option C.
Simplified Boolean Equation
Based on the grouping strategy focusing on the terms $\overline{x}z$ and $x\overline{z}$, the simplified Boolean equation is:
\(F = \overline{x}z + x\overline{z}\)
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?
The simplified form of a Boolean equation $(A\overline{B} + A\overline{B}C + AC)(\overline{A}\overline{C} + \overline{B})$ is :
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.
