Final answer:
The equation (A X A) (B X B) = (A | B) X AB is not true for any two sets A and B.
Step-by-step explanation:
To determine if the equation (A X A) (B X B) = (A | B) X AB holds for any two sets A and B, we can expand both sides of the equation and compare them. Let's start by expanding the left side:
(A X A) (B X B) = a1, a2 ∈ A b1, b2 ∈ B
By applying the definition of Cartesian product, we can further simplify it to:
(A X A) (B X B) = (a1, a2, b1, b2)
Now, let's expand the right side of the equation:
(A | B) X AB = c ∈ (A
Using the definitions of union and intersection, we can rewrite it as:
(A | B) X AB = (c, d)
By applying the definition of Cartesian product again, we can rewrite it as:
(A | B) X AB = (c ∈ A or c ∈ B), d, e ∈ A and (d, e) ∈ B
Comparing the expanded versions of both sides of the equation, we can see that they are not equal. Therefore, the equation (A X A) (B X B) = (A | B) X AB is not true for any two sets A and B.