How can the intersection property of two Cartesian products be understood graphically?

Conditions

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

Reasoning, step by step

1. Represent $A \times B$ as a red rectangle with width A and height B.
2. Represent $C \times D$ as a green rectangle with width C and height D.
3. Identify the overlapping region of the two rectangles.
4. Observe that the width of the overlapping region is A ∩ C.
5. Observe that the height of the overlapping region is B ∩ D.
6. Conclude that the overlapping region represents (A ∩ C) × (B ∩ D).

Example

The video shows two overlapping rectangles. The red one has width A and height B, and the green one has width C and height D. Their purple overlapping region has width A ∩ C and height B ∩ D, visually verifying the formula.

Common misconceptions

• Thinking that the intersection of two rectangles is always a rectangle.
• Confusing the width and height assignments of the rectangles.
• Assuming that the overlapping region's area is the sum of the areas of the two rectangles.

Watch the explanation

Explore next

• Does ($A \times B$) ∪ ($C \times D$) equal (A ∪ C) × (B ∪ D)?
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