Hall's Marriage Theorem/Explanation

Hall's Marriage Theorem: Explanation

This Hall's Marriage Theorem is so called for the following reason:

Let $I$ be a set of women.

Suppose that each woman $k$ is romantically interested in a finite set $S_k$ of men.

Suppose also that:

each woman would like to marry exactly one of these men

and:

each man in $\ds \bigcup_{k \mathop \in I} S_k$ would like to marry at most one woman in $I$.

Then this theorem gives a condition under which it is possible to match every woman to a man.