(A X B) U (C X D) ⊆ (A U C) X (B U D) <- that's a theorem.. i don't see it being true in this case, unless I'm doing something wrong?
A,B,C,D are sets with the following values:
A = { 1, 2, 3, 4}
B = { 2, 1 , 4}
C = { 4, 2 , 3}
D = {1, 5, 7}
A x B = { (1,2), (1,1), (1,4), (2,2), (2,1), (2,4), (3,2), (3,1), (3,4), (4,2), (4,1), (4,4) }
C x D = { (4,1), (4,5), (4,7), (2,1), (2,5), (2,7), (3,1), (3,5), (3,7) }
(A x B) U (C x D) = { (1,2), (1,1), (1,4) , (2,2), (2,4), (3,2), (3,4), (4,2), (4,4), (4,5), (4,7), (2,5),
(2,7), (3,5), (3,7) }
------------
A U C = { 1 }
B U D = { 2, 4, 5, 7 }
(A U C) X (B U D) = { (1,2), (1,4), (1,5), (1,7) }