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 depicted wiring contains ambiguities. We will proceed assuming a standard interpretation of MUX symbols and connections.Step 1: Analyze MUX 1.

The select input, connected to signal B, is the bottom input.
When select=0, the data input is C.
When select=1, the data input is \(\overline{C}\).
Therefore, the output equation for MUX 1, \(O_1\), is: \(O_1 = \overline{B} \cdot C + B \cdot \overline{C} = B \oplus C\).

Step 2: Analyze MUX 2.

The select input is connected to signal A.
When select=0, the data input is 0.
When select=1, the data input is \(O_1\).
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 differs from all provided options, suggesting a potential error in the original diagram or answer choices. Examining option B, \(AB\overline{C} + A\overline{B}\overline{C}\), reveals a strong similarity to our derived expression. It is likely that option B contains a typographical error. Assuming a standard circuit interpretation, our derived expression is accurate.