Exercise 3: Reusing a Majority Circuit as Full Adder Carry Logic — Possible Solution
====================================================================

THE FINDING THIS CHAPTER ESTABLISHED
------------------------------
This chapter proved directly (checked for all 8 combinations of
x, y, cin) that a full adder's COUT logic and the 3-input majority
function are exactly the same function - COUT is 1 in precisely
the same cases majority(x,y,cin) is 1. This isn't an approximate
resemblance; the two truth tables are identical.

WHY THIS MEANS THE CIRCUITS ARE INTERCHANGEABLE
------------------------------
A circuit is fully defined by its truth table - two circuits that
produce identical outputs for every possible input combination are
functionally indistinguishable from the outside, regardless of how
each one happens to be internally wired. Since the majority
circuit's own truth table and the full adder's COUT truth table are
identical, any circuit that correctly implements one automatically,
and exactly, implements the other. There is no additional
verification needed beyond confirming the truth tables match, which
this chapter's own exhaustive 8-case check already did.

THE PRACTICAL PAYOFF
------------------------------
A chip designer who already has a majority-function circuit that's
been tested and verified doesn't need to design, build, or
independently re-verify a new carry-out circuit for a full adder -
they can wire the exact same, already-proven component in with its
inputs simply relabeled x, y, cin instead of whatever they were
called in the original majority-circuit design. This is a real,
practical form of reuse: verification work done once (in this
chapter's own case, spanning Chapters 4 through 6) pays off again
in a completely different context (adder design), because the
underlying Boolean function is identical.

WHY THIS WORKS AS AN ANSWER
------------------------------
The explanation grounds the reuse claim in the specific, already-
verified fact this chapter established (matching truth tables,
checked exhaustively) rather than a vague "they're similar" claim,
and explains precisely why matching truth tables is sufficient
justification for full interchangeability, not just a loose
practical convenience.