CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-02

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.

A finite family has an SDR if and only if every subfamily has a union at least as large as its index set

Statement

Let $(A_{x})_{x \in X}$ have finite index set $X$ and finite union $U : = ⋃_{x \in X} A_{x}$ . It has an SDR if and only if $∣ ⋃_{x \in S} A_{x} ∣ \geq ∣ S ∣ (S \subseteq X) .$

Facts & Assumptions

Given: A family $(A_{x})_{x \in X}$ with finite $X$ and finite union $U$ , and its tagged incidence graph with parts $X_{L}, U_{R}$ .

[F1]

The tagged incidence graph is finite and bipartite, its left-neighbourhood of $S_{L}$ is the tagged copy of $⋃_{x \in S} A_{x}$ , and an SDR is an injection choosing one adjacent right tag for each left tag (Bipartite neighbourhoods, Hall's condition and systems of distinct representatives).

[L1]

A finite bipartite graph has a matching saturating its left part exactly when Hall's condition holds (Hall's marriage theorem for a finite bipartite graph).

Proof

technique · direct

1.1

The displayed union inequality is exactly Hall's condition for the finite tagged incidence graph.

1.2

By [F1] the tagged incidence graph is finite, so [L1] makes that condition equivalent to a matching that saturates $X_{L}$ .

F1L1

1.3

Such a matching assigns each $x$ the underlying element of its unique matched right tag, and conversely an SDR gives those pairwise disjoint tagged matching edges.

2.1

Combining steps 1.1--1.3 proves the stated equivalence.

step 1.1step 1.2step 1.3∎

Depends on

Used by

A three-set family with an explicit system of distinct representatives Example

Dependency tree · two levels

9 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

F. Gotti, Matching and Hall's Theorem (standard reference, not scraped)