Pigeonhole Principle
A function from one finite set to a smaller finite set cannot be one-to-one: There must be at least two elements in the domain that have the same image in the co-domain.
Intuition
Here is an intuitive example. Lets say you have a function $B$ that maps people to their month of birth. I propose that $B$ is not a function, if there are more than 12 people in it's domain.
Why is this?
Well how could it be a function if two elements map to the same value; that would constitute a fail of the vertical line test, a preliminary to a function.