Without any additional circuitry, an 8 : 1 MUX can be used to obtaina)some but not all Boolean functions of 3 variablesb)all functions of 3 variables, but none of 4 variablesc)all functions of 3 variables and some, but not all of 4 variablesd)all functions of 4 variablesCorrect answer is option 'D'. Can you explain this answer?

Question

Answer

A 2ⁿ : 1 MUX can implement all logic functions of (n + 1) variable without any additional circuitry.

Here, n = 3.

Thus, an 8 : 1 MUX can implement all logic functions of 4 variables.

Answer

Overview:

An 8:1 MUX (multiplexer) is a digital circuit that can select one input from eight different sources and route it to the output. Without any additional circuitry, it can be used to implement all functions of 4 variables.

Explanation:

To understand why an 8:1 MUX can implement all functions of 4 variables, let's first understand its basic operation.

8:1 Multiplexer:

- An 8:1 MUX has 8 input lines (I0, I1, ..., I7), 3 select lines (S0, S1, S2), and 1 output line (Y).
- The select lines determine which input line is routed to the output. The binary value of the select lines represents the index of the selected input line.
- For example, if S2S1S0 = 010, then input line I2 will be selected and routed to the output.

Implementing Functions of 3 Variables:

- When dealing with 3 variables (A, B, C), the remaining 4th variable is considered as a constant (either 0 or 1).
- The 8 input lines of the 8:1 MUX can be used to represent all possible combinations of the 3 variables (A=0, B=0, C=0), (A=0, B=0, C=1), ..., (A=1, B=1, C=1).
- By setting the select lines appropriately, we can map each input combination to the desired output value.
- Therefore, an 8:1 MUX can implement all functions of 3 variables.

Implementing Functions of 4 Variables:

- To implement functions of 4 variables (A, B, C, D), we can use two 8:1 MUXes and combine their outputs.
- The first 8:1 MUX can be used to select the output based on the values of A, B, and C.
- The second 8:1 MUX can be used to select the output of the first 8:1 MUX based on the value of D.
- By connecting the inputs and select lines appropriately, we can obtain the desired output for any combination of A, B, C, and D.
- Therefore, an 8:1 MUX can implement all functions of 4 variables.

Conclusion:

Without any additional circuitry, an 8:1 MUX can be used to implement all functions of 4 variables. This is achieved by using two 8:1 MUXes and combining their outputs. The first 8:1 MUX selects the output based on the values of A, B, and C, while the second 8:1 MUX selects the output of the first 8:1 MUX based on the value of D.

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