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I tried finding stuff online but I couldn't about partial matchings.
Definition we're given: A partial matching of $\mathcal{A}$ is a matching of some subfamily $\mathcal{B}\subseteq \mathcal{A}$.

A matching is defined:
A matching of $\mathcal{A}$ is a set of distinct elements $\{a_i\}_{i\in I}$ with $a_i \in A_i$ for all $i\in I$ where $\mathcal{A} = \{A_i\}_{i\in I}$.

Let $\mathcal{I}(\mathcal{A})$ be a family of partial matchings of $\mathcal{A}$.
Prove that if $A \in\mathcal{I}(\mathcal{A})$ and $B\subseteq A$, then $B\in\mathcal{I}(\mathcal{A})$.

I'm not sure where to start really and I tried using Hall's theorem for it to no avail.

OneGapLater
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  • If you say 'Let $\mathcal I(\mathcal A) $ be the set of all partial matchings', the proposition would trivially hold. Otherwise it seems to presume some further condition on the 'family'. – Berci Jun 18 '18 at 07:11
  • Oh sorry I didn't know of a difference. The notes say (verbatim): "Let $\mathcal{A} = {A_i}_{i\in I}$ be a finite family of subsets on a finite set $E$ and let $\mathcal{I}(\mathcal{A})$ be the family of partial matchings on $\mathcal{A}$. – OneGapLater Jun 18 '18 at 08:24

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