This problem involves an 8-to-1 multiplexer where the selection inputs $S_2, S_1, S_0$ are connected to $A, B, C$, respectively, and the data inputs $I_0$ to $I_7$ are given specific values based on the problem.

Problem Breakdown:

1. Inputs:

   • Selection lines: $S_2 = A$, $S_1 = B$, $S_0 = C$.
   • Data inputs: $I_1 = I_2 = 0, \quad I_3 = I_7 = 1, \quad I_4 = I_5 = D, \quad I_0 = I_6 = \overline{D}.$

2. Multiplexer Function:

   • The output of an $8 \times 1$ multiplexer is determined by the values of the selection lines $S_2, S_1, S_0$, which select one of the inputs $I_0$ to $I_7$.

3. Output Expression:

   • Based on the values of $S_2, S_1, S_0$, the corresponding input line is selected, and the output $Y$ is equal to that input.

Output Equation:

The output $Y$ can be expressed in terms of the selection inputs $A, B, C$ (or $S_2, S_1, S_0$):

$Y = I_0 \cdot \overline{A} \cdot \overline{B} \cdot \overline{C} + I_1 \cdot \overline{A} \cdot \overline{B} \cdot C + I_2 \cdot \overline{A} \cdot B \cdot \overline{C} + \dots + I_7 \cdot A \cdot B \cdot C.$

Substituting the values of $I_0$ to $I_7$:

$Y = \overline{D} \cdot \overline{A} \cdot \overline{B} \cdot \overline{C} + 0 \cdot \overline{A} \cdot \overline{B} \cdot C + 0 \cdot \overline{A} \cdot B \cdot \overline{C} + 1 \cdot \overline{A} \cdot B \cdot C + D \cdot A \cdot \overline{B} \cdot \overline{C} + D \cdot A \cdot \overline{B} \cdot C + \overline{D} \cdot A \cdot B \cdot \overline{C} + 1 \cdot A \cdot B \cdot C.$

Simplifying, $Y$ becomes:

$Y = \overline{D} \cdot \overline{A} \cdot \overline{B} \cdot \overline{C} + \overline{A} \cdot B \cdot C + D \cdot A \cdot \overline{B} + \overline{D} \cdot A \cdot B \cdot \overline{C} + A \cdot B \cdot C.$

Do you want step-by-step simplification for the final equation or more clarification?

5 Related Questions:

1. How does a multiplexer work in digital circuits?
2. What are the advantages of using multiplexers in logic design?
3. How can the Boolean expression for a multiplexer be further simplified using Karnaugh maps?
4. What is the role of the selection inputs $S_2, S_1, S_0$ in determining the output?
5. How would the output change if the data input values were different?

Tip:

When solving multiplexer problems, always write the selection logic and data inputs clearly, as this ensures correct substitution and simplifies Boolean expression derivation.