How can the distributive law of Cartesian product over union be understood graphically?

Conditions

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

Reasoning, step by step

1. Represent set C as the height of a rectangle.
2. Represent the union A ∪ B as the total width of the rectangle.
3. Divide the total width into two segments representing A and B.
4. Calculate the area of the left sub-rectangle as $A \times C$.
5. Calculate the area of the right sub-rectangle as $B \times C$.
6. Observe that the total area (A ∪ B) × C equals the sum of the two sub-areas ($A \times C$) ∪ ($B \times C$).

Example

The video shows a large rectangle with height C and base divided into segments A and B. The interior is split into two smaller rectangles labeled $A \times C$ and $B \times C$, visually verifying that the total area equals the sum of the parts.

Common misconceptions

• Thinking that the geometric model proves the algebraic identity rigorously without relying on the definition of Cartesian product.
• Confusing the width of the rectangle with the set C.
• Assuming that the union of two rectangles is always a single rectangle.

Watch the explanation

Explore next

• What is the intersection of two Cartesian products?
• Does ($A \times B$) ∪ ($C \times D$) equal (A ∪ C) × (B ∪ D)?
• What distributive laws does the Cartesian product satisfy with respect to union and intersection?