Which one of the following circuits implements the Boolean function given below?
\[ f(x,y,z) = m_0 + m_1 + m_3 + m_4 + m_5 + m_6, \] where $m_i$ is the $i^{\text{th}}$ minterm.

Question

Suppose we want to design a synchronous circuit that processes a string of 0's and 1's. Given a string, it produces another string by replacing the first 1 in any subsequence of consecutive 1's by a 0. Consider the following example.
Input sequence: 00100011000011100
Output sequence: 00000001000001100
A Mealy Machine is a state machine where both the next state and the output are functions of the present state and the current input. The above mentioned circuit can be designed as a two-state Mealy machine. The states in the Mealy machine can be represented using Boolean values 0 and 1. We denote the current state, the next state, the next incoming bit, and the output bit of the Mealy machine by the variables $s$, $t$, $b$ and $y$ respectively. Assume the initial state of the Mealy machine is 0. What are the Boolean expressions corresponding to $t$ and $y$ in terms of $s$ and $b$?

Answer 1

The given Boolean function is:

$f(x,y,z) = m_0 + m_1 + m_3 + m_4 + m_5 + m_6$

This represents the minterms of the function where the expression is true. Let's identify these minterms:

$m_0: \bar{x}\bar{y}\bar{z}$
$m_1: \bar{x}\bar{y}z$
$m_3: \bar{x}yz$
$m_4: x\bar{y}\bar{z}$
$m_5: x\bar{y}z$
$m_6: xy\bar{z}$

Next, we'll use a 4x1 multiplexer to implement this function. In a 4x1 MUX,

we choose two selection lines and use the input combinations to enable the required minterms.

Let's choose $y$ and $z$ as the selection lines:

When $y=0, z=0$: The output should be $1$, which corresponds to $m_0$
When $y=0, z=1$: The output should be $1$, which corresponds to $m_1$
When $y=1, z=0$: The output should be $1$, which corresponds to $m_6$
When $y=1, z=1$: The output should be $x'$ for $m_3$

Therefore, the MUX inputs should be configured as follows:

Input $0$: $1$
Input $1$: $1$
Input $2$: $1$
Input $3$: $x'$

This configuration matches the option A, as shown in the diagram:

Thus, the correct answer is Option A.

Which one of the following circuits implements the Boolean function given below?
\[ f(x,y,z) = m_0 + m_1 + m_3 + m_4 + m_5 + m_6, \] where \(m_i\) is the \(i^{\text{th}}\) minterm.

Show Hint

The Correct Option is A

Solution and Explanation

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Which one of the following circuits implements the Boolean function given below? \[ f(x,y,z) = m_0 + m_1 + m_3 + m_4 + m_5 + m_6, \] where \(m_i\) is the \(i^{\text{th}}\) minterm.