The Partition Principle states that, whenever there is a surjection from a set 
onto a set 
, there is an injection from
into 
.
The injection is not required to be a right inverse
of the given surjection. A right inverse would have
to choose one member of each fiber, a stronger requirement tied to the axiom
of choice.
Partition Principle
See also
Axiom of Choice, Injection, Surjection, Zermelo-Fraenkel Set TheoryExplore with Wolfram|Alpha
References
Blass, A. and Kulshreshtha, D. "A Gentle Introduction to the Axiom of Choice." Math. Intelligencer, 2026. https://doi.org/10.1007/s00283-026-10531-4.Cite this as:
Weisstein, Eric W. "Partition Principle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PartitionPrinciple.html