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The distributive law (A ∪ B) × $C = (A \times C)$ ∪ ($B \times C$) can be understood by treating sets as lengths of line segments and Cartesian products as areas of rectangles. The area of a large rectangle with width A ∪ B and height C equals the sum of the areas of two smaller rectangles with widths A and B, and height C.

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

The distributive law (A ∪ B) × $C = (A \times C)$ ∪ ($B \times C$) can be understood by treating sets as lengths of line segments and Cartesian products as areas of rectangles. The area of a large rectangle with width A ∪ B and height C equals the sum of the areas of two smaller rectangles with widths A and B, and height C.

Conditions: A, B, and C 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.

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 Cartesian product distributes over both union and intersection from both the left and the right. Specifically, (A ∪ B) × $C = (A \times C)$ ∪ ($B \times C$), (A ∩ B) × $C = (A \times C)$ ∩ ($B \times C$), A × (B ∪ C) = ($A \times B$) ∪ ($A \times C$), and A × (B ∩ C) = ($A \times B$) ∩ ($A \times C$).

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

The Cartesian product distributes over both union and intersection from both the left and the right. Specifically, (A ∪ B) × $C = (A \times C)$ ∪ ($B \times C$), (A ∩ B) × $C = (A \times C)$ ∩ ($B \times C$), A × (B ∪ C) = ($A \times B$) ∪ ($A \times C$), and A × (B ∩ C) = ($A \times B$) ∩ ($A \times C$).

Conditions: A, B, and C 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.

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.