The document describes a pairing between two infinite sets A and B where each element of A is paired with a unique element of B, but some elements of B are unpaired. This implies the cardinalities of A and B are not equal since the pairing is not surjective, with elements of B left unpaired. If A and B were finite sets instead of infinite, then the cardinalities could be equal if the number of unpaired elements of B is also finite.

1. Suppose you have two infinite sets A and B, and you are shown a pairing in which each element
from A is associated with exactly one element from B and no two elements of A are paired up
with the same element of B. In this pairing, however, there are elements of b that are not paired
up with elements of A.
Does this imply that the cardinality A and the cardinality of B are not equal? Carefully explain
why or why not.
Suppose we replace the word "infinite'' with "finite" would that change your answer?
Explain.
Solution
I believe that in order for the cardinality to be the same there must exist a mapping
that is not only one-to-one (injective) but also onto (surjective). The pairing you described
between A and B is not surjective since there exist elements in B that are not paired with A.