The Cartesian product of two sets X and Y is the set
That is, it is the set of all ordered pairs where the first coordinate comes from X and the second coordinate comes from Y.
A relation between two sets X and Y is any subset of
. That is, it is any subset of ordered pairs, where the
first coordinates are drawn from X and the second coordinates are
drawn from Y.
The domain of a relation S is the set of first coordinates
in the relation; i.e.,
The range of a relation S is the set of second coordinates in the relation; i.e.,
Here are my high school teacher's definitions: A ficklepicker of a relation is a first coordinate that appears in more than one ordered pair of the relation. A function is a relation with no ficklepickers.
If f is a function, then we write, for example, 
to mean
that the ordered pair 
is in the function.
When we write 
is a function, we mean that the
domain of f is the entire set X, and the range of f is contained
in (but not necessarily equal to) Y.
A function is one-to-one or an injection
if no element of the range is paired with more than one element of X.
A function is onto or a surjection if
its range is all of Y.
A function is one-to-one and onto
or a bijection if it is both one-to-one and onto.
If is a bijection and X and Y are both
finite sets, then we quite naturally say that X and Y are the same
size.
Try to find a bijection between the set 
of natural numbers and the set
of integers. Do you feel comfortable saying that and are
the same size?