König's theoremThere is also a proposition in graph theory called König's lemma.
In set theory, König's theorem states that if I is a set and mi and ni are cardinal numbers for every i in I, and
here is the disjoint union
of the sets ni
; and the product is the cartesian product
; we can similarly state it for arbitrary sets (not necessarily cardinal numbers) by replacing < by strictly less than in cardinality,
i.e. there is an injective function
but not one going the other way. The union involved need not be disjoint (a non-disjoint union can't be any bigger than the disjoint version, anyway).
(Of course this is trivial if the cardinal numbers mi and ni are finite and the index set I is finite. If I is empty, then the left sum is the empty sum and therefore 0, while the right hand product is the empty product and therefore 1).