Explore identities through matched incidence counts and combinatorial arguments. Test multiple core proof models instantly. Turn abstract counting ideas into verified numerical evidence today.
Choose a proof model, enter its parameters, and compare two independent counts of the same combinatorial object.
| Proof model | Input snapshot | Left count | Right count | Result |
|---|---|---|---|---|
| Handshake lemma | Degrees = 3, 3, 2, 2 and E = 5 | 10 | 10 | Verified |
| Committee incidence | N = 12, r = 3, C = 9, s = 4 | 36 | 36 | Verified |
| Committee with leader | n = 8, k = 3 | 168 | 168 | Verified |
| Bipartite incidence | |A| = 6, dA = 4, |B| = 8, dB = 3 | 24 | 24 | Verified |
Handshake lemma: ∑deg(v) = 2E. Count vertex-edge incidences by summing degrees, or count two endpoints for every edge.
Committee incidence: N × r = C × s. Count person-committee memberships from the people side and from the committee side.
Committee with leader identity: C(n,k) × k = n × C(n−1,k−1). Count ordered pairs of a committee and a distinguished leader in two ways.
Bipartite incidence: |A| × dA = |B| × dB. Count all edges once from partition A and again from partition B.
A double counting proof shows two expressions are equal because they count the same collection of objects from two different viewpoints. It is common in combinatorics, graph theory, and incidence arguments.
The left side and right side represent two counting strategies. When both totals match, the identity is numerically verified for the chosen input values, supporting the proof idea.
A mismatch usually means the inputs do not satisfy the stated combinatorial structure. It can also reveal inconsistent graph data, incorrect committee parameters, or a mistaken interpretation of the counting object.
Yes. The committee-with-leader mode verifies the classic identity C(n,k) × k = n × C(n−1,k−1), which is a standard double counting argument in discrete mathematics.
It counts the same set of vertex-edge incidences twice. Summing degrees counts incidences by vertices, while 2E counts each edge through its two endpoints.
Yes. These proof models describe discrete objects such as vertices, edges, committees, and chosen leaders. Those quantities are naturally integer counts rather than fractional values.
They contain the active proof model, the identity tested, the current inputs, both counts, the difference, and the final verdict. They are useful for worksheets, reports, and lectures.
Matching totals for one input set gives numerical confirmation. A full mathematical proof also explains clearly why both expressions always count the same underlying objects for all valid inputs.
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.