What is the intersection of two Cartesian products?

Conditions

• A, B, C, and D are arbitrary sets.

Reasoning, step by step

1. Identify the two Cartesian products $A \times B$ and $C \times D$.
2. Find the intersection of the first components: A ∩ C.
3. Find the intersection of the second components: B ∩ D.
4. Form the new Cartesian product (A ∩ C) × (B ∩ D).
5. Conclude that ($A \times B$) ∩ ($C \times D$) = (A ∩ C) × (B ∩ D).

Example

The video displays the formula ($A \times B$) ∩ ($C \times D$) = (A ∩ C) × (B ∩ D) and illustrates it with two overlapping rectangles, where the purple overlapping region has width A ∩ C and height B ∩ D.

Common misconceptions

• Believing that the intersection of two Cartesian products is always empty.
• Thinking that ($A \times B$) ∩ ($C \times D$) = (A ∩ D) × (B ∩ C).
• Assuming that the intersection property holds for union as well.

Watch the explanation

Explore next

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