***The following definitions are quite important, and we will use them for the remainder of
this class. Use them with precision enough so that you have them memorized.
I. Cartesian Products, Relations, and Functions
A. Definition: Cartesian Product of A and B
The Cartesian product A x B = {(a,b)| a € A ^ b € B}
B. Definition: Relation on A x B
A relation D x C on A x B is any subset of A x B, where D is called the domain of the relation and
C is called the codomain of the relation.
C. Definition: Function from D to C
Let D and C be sets. A function f from D to (into) C is a relation on D x C such that
i) For any x € D, there exists a y € C such that (x,y) € f. (Here we say y = f(x).) (Existence
or everywhere defined property)
ii) If (s,t) € f and (s,y) € f then t = y. (Uniqueness or well-defined property)
D. Definition: Range of a function
Let f be a function from D to C. The range of f is the set R = {y| y € C such that there is an x in D
for which (x,y) € f}.
E. Definition: one-to-one function (1-1)
Let f be a function from D to C. f is 1-1 <=> whenever (s,t) and (s',t) € f, then s = s'.
F. Definition: Onto function
Let f be a function from D to C. F is onto C <=> whenever y € C, there exists an x € D such that
(x,y) € f.
I. Fake Quiz Problems
1. Suppose D = {a,b,c,d,e} and C = (1,2,3,4,5}.
2. How many ordered pairs are in the relation D x C (and why)?
3. How many possible relations are there on D and C?
4. Which if any of the following are functions with domain D and codomain C? (State your
reasoning here.)
a) {(a,1), (b,2), (c,4), (d,3), (e,5)}
b) {(a,1), (b,1), (c,1), (d,1), (e,1)}
c) {(a,1), (b,1), (c,1), (d,1), (e,1) (f,1)}
d) {(a,1), (a,2), (a,4), (a,3), (a,5)}
e) {(a,1), (b,2), (b,4), (d,3), (e,5)}
f) {(a,1), (b,2), (c,4), (d,3)}
g) {(a,1), (e,5)}