Exercise 2: Gate Count for xy + x'y, Unsimplified vs. Simplified — Possible Solution
====================================================================

GIVEN
------------------------------
Unsimplified: f(x,y) = xy + x'y
Simplified (from Chapter 5 Exercise 1): f(x,y) = y

STEP 1: DESCRIBE THE UNSIMPLIFIED CIRCUIT, GATE BY GATE
------------------------------
- One NOT gate, producing x' from x
- One AND gate, computing xy (from the raw inputs x and y)
- One AND gate, computing x'y (from x' and y)
- One OR gate, combining xy and x'y into the final output

Gate count: 1 NOT + 2 AND + 1 OR = 4 gates total

STEP 2: DESCRIBE THE SIMPLIFIED CIRCUIT
------------------------------
The simplified expression is just y - the output IS one of the
original inputs, wired straight through with no gates at all
needed in between.

Gate count: 0 gates

RESULT
------------------------------
Unsimplified circuit: 4 gates (1 NOT, 2 AND, 1 OR)
Simplified circuit: 0 gates (a direct wire from input y to output)

This is the most extreme possible case of Chapter 6's own
gate-count-reduction principle - simplification didn't just make
the circuit smaller, it eliminated the need for a circuit at all.

WHY THIS WORKS AS AN ANSWER
------------------------------
The unsimplified circuit's gate count is built up explicitly, gate
by gate, matching how the raw Boolean expression xy + x'y would
actually need to be wired (including the NOT gate its own x' term
requires), and the comparison against the zero-gate simplified
version makes explicit just how large a reduction is possible when
an entire variable drops out during simplification.