What distributive laws does the Cartesian product satisfy with respect to union and intersection?

Conditions

• A, B, and C are arbitrary sets.

Reasoning, step by step

1. Recall the definition of the Cartesian product.
2. Apply the distributive property for union on the left side.
3. Apply the distributive property for intersection on the left side.
4. Apply the distributive property for union on the right side.
5. Apply the distributive property for intersection on the right side.

Example

The video lists four formulas: (1) (A ∪ B) × $C = (A \times C)$ ∪ ($B \times C$); (2) (A ∩ B) × $C = (A \times C)$ ∩ ($B \times C$); (3) A × (B ∪ C) = ($A \times B$) ∪ ($A \times C$); (4) A × (B ∩ C) = ($A \times B$) ∩ ($A \times C$).

Common misconceptions

• Believing that the Cartesian product only distributes over union.
• Thinking that the distributive laws only apply from the left side.
• Confusing the distributive laws with the intersection property of two Cartesian products.

Watch the explanation

Explore next

• How can the distributive law of Cartesian product over union be understood graphically?
• What is the intersection of two Cartesian products?
• Does ($A \times B$) ∪ ($C \times D$) equal (A ∪ C) × (B ∪ D)?