Consider the circuit in below figure. f implements

To determine which boolean function the 4-to-1 Multiplexer (Mux) implements, we can derive its output equation based on the select lines and input lines.

Step-by-Step Derivation

A 4-to-1 multiplexer selects one of its four data inputs (I0, I1, I2, I3) based on the binary combination of its select lines (S1, S0).

From the given circuit diagram:

• Select lines: S1 = A, S0 = B

• Data inputs:

  • I0 = C

  • I1 = ~C

  • I2 = ~C

  • I3 = C

The general characteristic equation for a 4-to-1 Mux is:

Substituting our specific values into the equation:

Simplifying the Expression

Let's group the terms to recognize standard logic gates:

1. Grouping the first and last terms, and the middle two terms:

   

2. Recognize the standard XOR and XNOR relations:

   

3. Substitute these back into the expression:

   

4. Let Y = A ⊕ B. The expression becomes:

   f = (~Y)C + Y(~C)

   This is the exact definition of an XOR operation between $Y$ and $C$:

   f = Y ⊕ C

5. Substitute Y = A ⊕ B back in:

   f = A ⊕ B ⊕ C