Exercise 1: Proving the Commutative Law for AND — Possible Solution
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GIVEN
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Prove xy = yx for all values of x and y, using a full truth table.

TRUTH TABLE, ALL 4 COMBINATIONS
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x=0, y=0: xy = 0.0 = 0. yx = 0.0 = 0. Match.
x=0, y=1: xy = 0.1 = 0. yx = 1.0 = 0. Match.
x=1, y=0: xy = 1.0 = 0. yx = 0.1 = 0. Match.
x=1, y=1: xy = 1.1 = 1. yx = 1.1 = 1. Match.

RESULT
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xy and yx produce identical results across all 4 possible input
combinations - since these are the only inputs a Boolean variable
pair can ever take, this is a complete proof, not just supporting
evidence. The commutative law xy = yx holds.

WHY THIS WORKS AS AN ANSWER
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All 4 combinations are checked explicitly rather than a
representative sample, matching this chapter's own stated proof
method - since there are only 4 possible (x,y) pairs total, checking
all of them genuinely does prove the law completely, not just make
it likely.