A Boolean function, $f(x,y,z)$ with $x$ as MSB and $z$ as LSB is realized by 4:1 multiplexer (MUX) with select lines, $S_1$ and $S_0$ ( $S_1$ is MSB, $S_0$ is LSB) and inputs, $I_0$, $I_1$, $I_2$, $I_3$ as shown in the Figure.
Which of the following options is the correct expression of $f(x,y,z)$?
To solve this problem, we need to understand how the Boolean function \(f(x, y, z)\) is realized using a 4:1 multiplexer (MUX).
The configuration given shows a 4:1 MUX with select lines \(S_1\) and \(S_0\), where \(S_1\) is the most significant bit (MSB) and \(S_0\) is the least significant bit (LSB).
- The inputs to the MUX are:
- \(I_0 = 1\)
- \(I_1 = 0\)
- \(I_2 = y\)
- \(I_3 = z\)
- The select lines are given as:
- \(S_1 = x\)
- \(S_0 = z\)
The output \(f(x, y, z)\) depends on the values of \(S_1\) and \(S_0\), which select one of the four inputs:
| \(S_1S_0\) | Selected Input |
|---|---|
| \(00\) | \(I_0 = 1\) |
| \(01\) | \(I_1 = 0\) |
| \(10\) | \(I_2 = y\) |
| \(11\) | \(I_3 = z\) |
We need to evaluate \(f(x, y, z)\) for different values of \(x\) and \(z\):
- When \(x = 0, z = 0\): \(f(0, y, 0) = I_0 = 1\)
- When \(x = 0, z = 1\): \(f(0, y, 1) = I_1 = 0\)
- When \(x = 1, z = 0\): \(f(1, y, 0) = I_2 = y\)
- When \(x = 1, z = 1\): \(f(1, y, 1) = I_3 = z\)
Therefore, the function is described as:
\(f(x, y, z) = (\bar{x} \bar{z}) \cdot 1 + (\bar{x} z) \cdot 0 + (x \bar{z}) \cdot y + (x z) \cdot z\)
Which simplifies to:
\(f(x, y, z) = (\bar{x} \bar{z}) + (x \bar{z}) y + (x z) z\)
Further simplification results in:
\(f(x, y, z) = \bar{z} + xy\)
This matches with the given option \(xy + \bar{z}\).
Thus, the correct solution is:
Correct Answer: \(xy + \bar{z}\)
Important Questions from Multiplexer
The device which changes from serial data to parallel data is:
Consider a logic gate circuit. with 8 input lines (D 0, D 1..... D 7) and 3 output lines (A 0, A 1, A 2) specified by following operations
A 2= D 4+ D 5+ D 6+ D 7
A 1= D 2+ D 3+ D 6+ D 7
A 0= D 1+ D 3+ D 5+ D 0
Where + indicates logical OR operation. This circuit is
Which of the following statements are true?
A. Frequency division multiplexing technique can be handled by digital circuits.
B. Time division multiplexing technique can be handled by analog circuits
C. Wavelength division multiplexing technique is used with optical fiber for combining two signals.
D. Frequency division multiplexing technique can be applied when the bandwidth of a link is greater than the bandwidth of the signals to be transmitted.
Choose the correct answer from the options given below :
Consider the sentence below.
There is a country that borders both India and Pakistan.
Which of the following logical expressions express the above sentence correctly when the predicate Country(x) represents that x is a country and Borders(x, y) represents that the countries x and y share the border?
A digital computer has a common bus system for 8 registers 16 bits each. How many multiplexers are required to implement common bus? What size of multiplexers is required?
