Understanding Cartesian Products

The Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B. This operation is fundamental in set theory and has applications in database operations, coordinate systems, and combinatorial mathematics.

Example Calculation

For sets A = {1, 2} and B = {x, y}, the Cartesian product is:

A × B = {(1, x), (1, y), (2, x), (2, y)}

Properties

The cardinality (size) of A × B is |A| × |B|
Cartesian products are non-commutative: A × B ≠ B × A (unless A = B)
Cartesian products can be extended to more than two sets

Frequently Asked Questions

What's the maximum number of sets I can compute?

This calculator supports up to 5 sets for performance reasons. The total number of tuples grows exponentially with each additional set.

How are empty sets handled?

The Cartesian product with any empty set results in an empty set, as there are no elements to form pairs with.

Can I use duplicate elements within a set?

Yes, but they will be treated as distinct elements unless you remove duplicates manually.

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