Question

Number of surjections into a set with 2 elements:

For every set A, |A| = n, n>1, how many surjective functions are there from A to B, where |B| = 2?

What I think:
A surjective function is basically any function from B to A. If n = 2, there are 2 surjective functions, if n=2, there are 2, n = 3, there are 6, if n = 4, there are 24, etc... (Pictures posted below)
2,6,24,120,720 follow the pattern n! so there are n! surjective functions.

What the answer was: "There are only 2 nonsurjective functions: the constants, so there are (2^n)-2 surjective functions"

Me: ">:o" Can anyone clarify this to me?

Answer 1

If n=2, there are 2 surjective functions:

If n = 3, there are 6 surjective functions:
.......................oh I think I noticed my mistake.....the one above isn't even a function REEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE RIP 33/200 points on my exam

Answer 2

My mistake was that I thought this was a function