Calculator Input
Choose a mode, enter values, and submit. Results appear above this form under the page header.
Example Data Table
| Mode | Example Input | Expected Output Summary |
|---|---|---|
| Finite sets | {a,b}; {1,2,3}; {x,y} | 12 tuples in A × B × C |
| Matrices | A = [[1,2],[3,4]], B = [[0,5],[6,7]] | 4 × 4 Kronecker product matrix |
| Orders of factors | 4, 6, 5 | Direct product order = 120 |
Formula Used
1) Cartesian product of finite sets
If the factors are A1, A2, ..., An, then the direct product contains all ordered tuples:
A1 × A2 × ... × An
The total number of tuples is:
|A1 × A2 × ... × An| = |A1| × |A2| × ... × |An|
2) Kronecker product of matrices
For matrices A = [aij] and B, the direct product used here is the Kronecker product:
A ⊗ B = [aijB]
If A is m × n and B is p × q, then A ⊗ B is (m·p) × (n·q).
3) Direct product order of finite factors
For finite factors G1, G2, ..., Gn, the order of the product is:
|G1 × G2 × ... × Gn| = |G1| × |G2| × ... × |Gn|
If each factor is cyclic and the orders are pairwise coprime, the overall product is cyclic.
How to Use This Calculator
- Select the direct product mode that matches your problem.
- Enter sets, matrices, or finite factor orders in the matching fields.
- Choose tuple limit or numeric precision where relevant.
- Press Calculate Direct Product to generate the result.
- Review the summary table, detailed output, and interpretation notes.
- Use the export buttons to save the result as CSV or PDF.
Frequently Asked Questions
1) What does this direct product calculator handle?
It handles Cartesian products of finite sets, Kronecker products of matrices, and order calculations for direct products of finite factors in algebra.
2) What format should I use for sets?
Write each component set with comma-separated elements. Separate different sets with semicolons. Example: {a,b}; {1,2,3}; {x,y}.
3) How are repeated elements treated in sets?
Repeated entries inside one set are removed before counting. This keeps the cardinality and tuple list consistent with standard finite-set interpretation.
4) What matrix product is used here?
The matrix mode uses the Kronecker product. Each entry of the first matrix multiplies the entire second matrix, creating a larger block matrix.
5) Why does the displayed tuple count stop early?
Large Cartesian products can contain many tuples. The display limit prevents extremely long pages while still reporting the full total cardinality.
6) What does pairwise coprime mean in the order mode?
It means every two distinct factor orders have greatest common divisor one. Under cyclic-factor assumptions, this supports a cyclic overall product.
7) Can I export the output for reports?
Yes. Use the CSV button for table-based data and the PDF button for a print-ready document containing the current result section.
8) Is this calculator suitable for learning and checking homework?
Yes. It shows intermediate structure summaries, dimension checks, tuple counts, and direct formulas, making it useful for study, verification, and teaching.