Question

The Boolean function f implemented in the figure using two input multiplexers is

• A. \(\overline{A}B\overline{C} + AB\overline{C}\)
• B. \(AB\overline{C} + A\overline{B}\overline{C}\)
• C. \(\overline{A}B\overline{C} + \overline{A}\overline{B}\overline{C}\)
• D. \(A B C + \overline{A} B C\)

Hint:

The output of a 2-to-1 MUX with select line S and data inputs \(I_0, I_1\) is always \(f = \overline{S} \cdot I_0 + S \cdot I_1\). Analyze cascaded MUX circuits by finding the output of the first stage and using it as an input to the next.

Answer 1

The Correct Option is D

Note: The wiring in the diagram is ambiguous. We will proceed with the most standard interpretation of the MUX symbols and connections. Step 1: Analyze the first multiplexer (MUX 1).

The select line is the bottom input, which is connected to signal B.
The data input for select=0 is C.
The data input for select=1 is \(\overline{C}\).
The output equation for this MUX, let's call it \(O_1\), is: \(O_1 = \overline{B} \cdot C + B \cdot \overline{C} = B \oplus C\).

Step 2: Analyze the second multiplexer (MUX 2).

The select line is connected to signal A.
The data input for select=0 is 0.
The data input for select=1 is the output of the first MUX, \(O_1\).
The final output equation f is: \(f = \overline{A} \cdot 0 + A \cdot O_1 = A \cdot O_1\).

Step 3: Substitute the expression for \(O_1\) into the equation for f. \[ f = A \cdot (B \oplus C) = A \cdot (\overline{B}C + B\overline{C}) \] \[ f = A\overline{B}C + AB\overline{C} \] This result does not match any of the provided options exactly as written, suggesting an error in the question's diagram or options. However, if we re-examine option B, which is \(AB\overline{C} + A\overline{B}\overline{C}\), it is very close to our derived expression. It seems likely there is a typo in the provided options. Based on a standard interpretation of the circuit, our derived expression is correct.