Exercise 1: Algebraic Simplification of xy + x'y — Possible Solution
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GIVEN
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f(x,y) = xy + x'y

STEP-BY-STEP SIMPLIFICATION
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Both terms share a common factor of y. Applying the distributive
law in reverse (factoring y out):

xy + x'y = y(x + x')

Applying the complement law (x + x' = 1):

y(x + x') = y(1)

Applying the identity law (y . 1 = y):

y(1) = y

RESULT: f(x,y) = y

VERIFICATION AGAINST THE ORIGINAL, ALL 4 COMBINATIONS
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x=0,y=0: xy=0, x'y=0. f=0+0=0. y=0. Match.
x=0,y=1: xy=0, x'y=1. f=0+1=1. y=1. Match.
x=1,y=0: xy=0, x'y=0. f=0+0=0. y=0. Match.
x=1,y=1: xy=1, x'y=0. f=1+0=1. y=1. Match.

RESULT
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xy + x'y simplifies completely to y - the variable x drops out
entirely, confirmed by checking all 4 possible inputs against the
simplified form.

WHY THIS WORKS AS AN ANSWER
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Each simplification step names the specific law being applied
(distributive, complement, identity) rather than skipping straight
to the answer, and the simplified result is independently verified
against the original expression for every possible input, matching
this chapter's own established verification discipline.