CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-07-31

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The complete graph on an $n$ -element vertex set has $(\binom{n}{2})$ edges

Statement

If $V$ is an $n$ -element set, then the complete graph $K_{V}$ has exactly $(\binom{n}{2})$ edges.

Facts & Assumptions

Given: A finite set $V$ with $∣ V ∣ = n$ .

[F1]

The edge set of $K_{V}$ is $[V]^{2}$ , the set of all two-element subsets of $V$ (Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

[L1]

An $n$ -element set has exactly $(\binom{n}{2})$ two-element subsets (A finite set with $n$ elements has exactly $(\binom{n}{2})$ two-element subsets, and $2 (\binom{n}{2}) = n (n - 1)$ ).

Proof

technique · direct

1.1

By [F1], $E (K_{V}) = [V]^{2}$ .

2.1

By [L1], $∣ [V]^{2} ∣ = (\binom{n}{2})$ , so step 1.1 gives $∣ E (K_{V}) ∣ = (\binom{n}{2})$ .

step 1.1L1∎

Depends on

Used by

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

R. Diestel, Graph Theory, Chapter 1 preview (standard reference, not scraped)

The complete graph on an nn-element vertex set has (n2)(2n​) edges