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?
A surjective map, also known as a surjection or onto function, is a function between two sets where every element in the output set is mapped to by at least one element in the input set. In other words, every element in the output set has a corresponding element in the input set.
A surjective map is a function that maps every element in the output set to by at least one element in the input set, while an injective map is a function that maps each element in the input set to a unique element in the output set. In other words, a surjective map covers the entire output set, while an injective map covers the entire input set.
A surjective map is typically written using function notation, with an arrow connecting the input set to the output set. For example, if the input set is denoted as X and the output set is denoted as Y, a surjective map can be written as f: X → Y.
Surjective maps are used to describe relationships between sets, where every element in the output set has a corresponding element in the input set. They are often used in topics such as algebra, topology, and analysis to describe functions that cover the entire output set.
Yes, a function can be both surjective and injective. This type of function is known as a bijective function. It is a one-to-one correspondence between the input and output sets, meaning that each element in the output set is mapped to by exactly one element in the input set, and vice versa.