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Cardinality Formula:
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The cardinality of a Cartesian product A × B is the number of ordered pairs where the first element comes from set A and the second element comes from set B. For finite sets, this equals the product of the cardinalities of A and B.
The calculator uses the cardinality formula:
Where:
Explanation: Each element of set A can be paired with each element of set B, resulting in the product of their cardinalities.
Details: Calculating the cardinality of Cartesian products is fundamental in set theory, combinatorics, database operations (SQL joins), and probability calculations where we need to determine the size of sample spaces.
Tips: Enter the number of elements in set A and set B as non-negative integers. The calculator will compute the number of ordered pairs in their Cartesian product.
Q1: What if one set is empty?
A: If either set A or set B is empty (cardinality 0), their Cartesian product will also be empty (cardinality 0).
Q2: Does order matter in Cartesian products?
A: Yes, Cartesian products consist of ordered pairs (a,b) where a ∈ A and b ∈ B. Generally, A × B ≠ B × A unless A = B.
Q3: How does this extend to more than two sets?
A: For n sets, the cardinality of their Cartesian product is the product of all their individual cardinalities: |A₁ × A₂ × ... × Aₙ| = |A₁| × |A₂| × ... × |Aₙ|
Q4: What's the difference between Cartesian product and power set?
A: The Cartesian product creates ordered pairs from elements of different sets, while the power set contains all subsets of a single set. Their cardinalities follow different formulas.
Q5: Can this calculator handle infinite sets?
A: No, this calculator is designed for finite sets only. Infinite set cardinalities require more advanced mathematical treatment.