Category:Hall's Marriage Theorem

This category contains pages concerning Hall's Marriage Theorem:

Finite Indexed Family of Finite Sets

Let $\SS = \family {S_k}_{k \mathop \in I}$ be a finite indexed family of finite sets.

For each $F \subseteq I$, let $\ds Y_F = \bigcup_{k \mathop \in F} S_k$.

Let $Y = Y_I$.

Then the following are equivalent:

$(1): \quad \SS$ satisfies the marriage condition: for each finite subset $F \subseteq I : \card F \le \card {Y_F}$.

$(2): \quad$ There exists an injection $f: I \to Y$ such that $\forall k \in I: \map f k \in S_k$.

General Indexed Family of Finite Sets

Let $\SS = \family {S_k}_{k \mathop \in I}$ be an indexed family of finite sets.

For each $F \subseteq I$, let $\ds Y_F = \bigcup_{k \mathop \in F} S_k$.

Let $Y = Y_I$.

Then the following are equivalent:

$(1): \quad \SS$ satisfies the marriage condition: for each finite subset $F \subseteq I : \card F \le \card {Y_F}$.

$(2): \quad$ There exists an injection $f: I \to Y$ such that $\forall k \in I: \map f k \in S_k$.