A = {0, 1}
B = {2, 3, 4}
C = {3, 5}
Then we have
A × (B U C)
= {0, 1} × {2, 3, 4, 5}
= {{0, 2}, {0, 3}, {0, 4}, {0, 5}, {1, 2}, {1, 3}, {1, 4}, {1, 5}}
and
(A × B) U (A × C)
= {{0, 2}, {0, 3}, {0, 4}, {1, 2}, {1, 3}, {1, 4}} U {{0, 3}, {0, 5}, {1, 3}, {1, 5}}
(underlined pairs are duplicates and common to both A × B and A × C)
= {{0, 2}, {0, 3}, {0, 4}, {0, 5}, {1, 2}, {1, 3}, {1, 4}, {1, 5}}
and these two sets are clearly identical.
