The discussion revolves around counting the number of surjective maps from a finite set S = {1,2,3,...,n} to itself. Participants are exploring the implications of surjectivity and the relationship between surjective and injective mappings in this context.
The discussion is active, with participants clarifying their understanding of surjective mappings and questioning assumptions about injectivity. There is acknowledgment of the relationship between the properties of the mappings and the finite nature of the set S.
Participants are grappling with the definitions and implications of surjective and injective mappings, particularly in the context of finite sets. The discussion highlights the importance of these definitions in determining the number of valid mappings.
battousai said:Homework Statement
Let S = {1,2,3,...,n}
How many surjective maps are there from S to S?The book's answer is n!
However, I thought that total number of surjective maps = n^n because 1-1 isn't required. Where am I wrong?
Arianwen said:1-1 is implicitly required, because each element of the domain can only map to one element of the range - so to map to every element of the range you'll need a 1-1 mapping.
LCKurtz said:But since S is finite, if the map wasn't 1-1 it couldn't be onto, could it?