The cartesian product of two sets is the set of all possible ordered pairs containing a member from each of those two sets. If set A contains 1, 2 and 3, and set B contains a, b and c, then the cartesian product of A and B would be {(1,a), (1,b), (1,c), (2,a), (2,b), (2,c), (3,a), (3,b), (3,c)}. No member of A or B is left out, and every member of A is paired with every member of B.
A relation is a subset of the cartesian product that pairs some of the members of one set with some of the members of another set. Some of the members of the first set may have more than one matching member from the second set, and vice versa. For example, if set A contains all the first names of students in a classroom, and set B contains all the last names, then a relation of the two sets would be the set of first and last names of the students. Some of the students may have the same first name, and some of the students might have the same last name.
A function is a type of relation in which each member of the first set has no more than one matching member of the second set. The above example would not be a function if any of the students had the same last name. A relation that paired the full names of the students with their student identification numbers would be a function, since none of the students would have the same identification number. The first set is called the domain of the function. The second set is called the co-domain.
Relations and functions are both subsets of the cartesian product of two sets. In both cases, some of the members of one set are paired with some of the members of another set. In both relations and functions, there may be some members of the second set that do not have a matching pair.