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The property ($A \times B$) ∩ ($C \times D$) = (A ∩ C) × (B ∩ D) can be understood by representing Cartesian products as rectangles. The intersection of two rectangles is a smaller rectangle whose width is the intersection of the widths of the original rectangles, and whose height is the intersection of their heights.

Conditions: A, B, C, and D are arbitrary sets.; The geometric analogy treats sets as lengths and Cartesian products as rectangular areas.

The property ($A \times B$) ∩ ($C \times D$) = (A ∩ C) × (B ∩ D) can be understood by representing Cartesian products as rectangles. The intersection of two rectangles is a smaller rectangle whose width is the intersection of the widths of the original rectangles, and whose height is the intersection of their heights.

Conditions: A, B, C, and D are arbitrary sets.; The geometric analogy treats sets as lengths and Cartesian products as rectangular areas.

No, ($A \times B$) ∪ ($C \times D$) generally does not equal (A ∪ C) × (B ∪ D). While the intersection of two Cartesian products is a Cartesian product, their union is typically an irregular shape (like an L-shape) that cannot be expressed as a single Cartesian product of two sets.

Conditions: A, B, C, and D are arbitrary sets.

No, ($A \times B$) ∪ ($C \times D$) generally does not equal (A ∪ C) × (B ∪ D). While the intersection of two Cartesian products is a Cartesian product, their union is typically an irregular shape (like an L-shape) that cannot be expressed as a single Cartesian product of two sets.

Conditions: A, B, C, and D are arbitrary sets.

The intersection of two Cartesian products ($A \times B$) and ($C \times D$) is equal to the Cartesian product of the intersections of their respective component sets, i.e., (A ∩ C) × (B ∩ D).

Conditions: A, B, C, and D are arbitrary sets.

The intersection of two Cartesian products ($A \times B$) and ($C \times D$) is equal to the Cartesian product of the intersections of their respective component sets, i.e., (A ∩ C) × (B ∩ D).

Conditions: A, B, C, and D are arbitrary sets.