CorollaryStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31

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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.
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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 $\binom n2$ 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 $\binom n2$ 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], $|[V]^2|=\binom n2$ , so step 1.1 gives $|E(K_V)|=\binom n2$ .

step 1.1L1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 53 results over 16 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

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

The complete graph on an nnn-element vertex set has (n2)\binom{n}{2}(2n​) edges