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

Conditions

• A, B, C, and D are arbitrary sets.

Reasoning, step by step

1. Recall the intersection property: ($A \times B$) ∩ ($C \times D$) = (A ∩ C) × (B ∩ D).
2. Attempt to analogize this property for union.
3. Observe the geometric representation of the union of two rectangles.
4. Note that the union of two overlapping or disjoint rectangles is not a single rectangle.
5. Conclude that the analogous formula for union does not hold.

Example

The video explicitly states: 'This diagram also shows that replacing intersection with union does not yield a similar valid theorem.' Geometrically, the union of two rectangles is generally not a regular rectangle.

Common misconceptions

• Assuming that all properties of intersection have analogous properties for union.
• Believing that the union of two Cartesian products is always a Cartesian product.
• Confusing the distributive laws with the intersection/union properties of two separate Cartesian products.

Watch the explanation

Explore next

• What is the intersection of two Cartesian products?
• How can the intersection property of two Cartesian products be understood graphically?
• What distributive laws does the Cartesian product satisfy with respect to union and intersection?