CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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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 $U:=\bigcup_{x\in X}A_x$ . It has an SDR if and only if $\left|\bigcup_{x\in S}A_x\right|\ge |S|\qquad(S\subseteq X).$

Facts & Assumptions

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

[F1]

The tagged incidence graph is finite and bipartite, its left-neighbourhood of $S_{\mathrm L}$ is the tagged copy of $\bigcup_{x\in S}A_x$ , 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 $X_{\mathrm L}$ .

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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 21 results over 13 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

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