Model images from algebraic maps quickly. Choose set mappings or modular rules. Clear outputs help you verify structure and learn confidently.
For a homomorphism f: A → B and a subset S ⊆ A, the image of S is:
For modular maps Zₙ → Zₘ with a multiplier k, we use:
Example: Z₈ → Z₆ with k = 3, subset S = {0,1,2,3}.
| x | f(x) = (3·x) mod 6 | Output |
|---|---|---|
| 0 | (3·0) mod 6 | 0 |
| 1 | (3·1) mod 6 | 3 |
| 2 | (3·2) mod 6 | 0 |
| 3 | (3·3) mod 6 | 3 |
The image set summarizes which outputs are actually reached by a map. When A has 8 elements and a subset S has 4 elements, the image can be as small as 1 element or as large as 4, depending on collisions. If two different inputs land on the same output, the image shrinks while the mapping table still records every input.
For f(x) = (k·x) mod m, outputs repeat with a period driven by m and k. Example: n = 8, m = 6, k = 3 gives f(S) = {0,3} for S = {0,1,2,3}, so the image size is 2 while the subset size is 4.
In pair mode, the calculator reports “mapped coverage” as a percentage of subset elements that have defined pairs. If S has 10 inputs but only 7 are mapped, coverage is 70%, and unmapped items appear with a dash in the table. This makes it easy to spot incomplete definitions before you export results for a report.
The bar chart counts how many times each value occurs among f(x) for x ∈ S. A tall bar indicates many collisions, which often signals a non-injective restriction on S. A flatter chart suggests outputs are more evenly spread. For instance, counts like {0:2, 3:2} imply a 50% collision rate for |S| = 4.
Start by loading the example, then adjust k or m and observe how the image changes. Try k = 1 to keep values aligned, then increase k to create wraps. If n = 12, m = 6, k = 2, the image cannot exceed 3 values. For pair mode, paste a mapping, set S, compute, and export CSV to keep a reproducible audit trail.
Useful metrics include |S|, |f(S)|, collision rate = 1 − |f(S)|/|S|, and the most frequent output in the chart. Across runs, compare image sizes for different subsets, or record how coverage changes as you complete missing pairs. When you save both CSV and PDF, you can attach the table plus metrics in a single workflow.
The image is the set of outputs produced by applying the map to a chosen input set. For a subset S, it is f(S) = { f(x) : x ∈ S }.
Different inputs can map to the same output. These collisions reduce the number of distinct outputs, so |f(S)| can be smaller than |S| even when every input is mapped.
It uses the rule f(x) = (k·x) mod m for x in {0,…,n−1}. The subset field lets you restrict which inputs are included in the image computation.
It is the percentage of subset elements that have a defined mapping pair. Unmapped inputs are listed in the table so you can complete the definition and recompute.
The chart shows frequencies of each output value among f(x) for x ∈ S. Higher bars indicate more collisions on that output, which often signals less injectivity on S.
The exports include the domain summary, subset summary, the image set, and the applied mapping table. CSV is best for reuse in spreadsheets; PDF is convenient for sharing a fixed report.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.