Consider the circuit shown below. The output of a 2:1 Mux is given by the…

Concept. A 2:1 multiplexer has two data inputs a and b and one select input c, and its output is ac' + bc. The select line does nothing more than choose which data input is passed through: when c = 0 the output equals a, and when c = 1 the output equals b. When two multiplexers are cascaded, the output of the earlier stage becomes a data input of the later stage, so the overall function is found by writing each stage's output with this same expression and then substituting the earlier one into the later one.

Application. Apply that expression to the two stages of the given circuit in order.

1. First stage: its data inputs are the constants a = 1 and b = 0, and its select input is c = x1. Substituting into ac' + bc gives g = 1·x1' + 0·x1 = x1'.

2. Second stage: its data input a is the first stage output g, its data input b is x1, and its select input is c = x2. Substituting into ac' + bc gives f = g·x2' + x1·x2.

3. Substitute g = x1' from the first stage: f = x1'x2' + x1x2, that is f = x1x2 + x1'x2'.

Cross-check. Evaluate the circuit directly for all four input combinations and compare with that expression.

Result. The circuit output is 1 exactly on the two rows where the inputs agree, so f = x1x2 + x1'x2', the equivalence (XNOR) of the two inputs. The complementary pattern x1'x2 + x1x2' is the exclusive-OR, which is 1 on the two rows where the inputs differ.