1. One-to-One and Onto, Inverse Functions
Lecture 32
Section 7.2
Wed, Mar 16, 2005

2. One-to-one Correspondences
A function f : A  B is a one-to-one correspondence if f is one-to-one and onto.
f has all four of the basic properties.
f establishes a ‘pairing” of the elements of A with the elements of B.

3. One-to-one Correspondences
One-to-one correspondences are very important because two sets are considered to have the same number of elements if there exists a one-to-one correspondence between them.

4. Example: One-to-one Correspondence
Are any of the following “functions” one-to-one correspondences?
f : Q  Q by f(a/b) = 2a/b.
g : Q  Q by g(a/b) = b/a.
h : Q  Q by h(a/b) = (a/b)2.
m : Q  Q by m(a/b) = a/b.
For the relation g, what if we replace Q with Q – {0}?

5. Inverse Relations Let R be a relation from A to B.
The inverse of R is the relation R–1 from B to A defined by the property that (x, y)  R–1 if and only if (y, x)  R.

6. Example: Inverse Relation
Let f : Q  Q by f(a/b) = 2a/b.
Describe f –1.
f –1(a/b) = a/2b.
Which of the four basic properties does f –1 have?

7. Example: Inverse Relation
Let g : Q  Q by g(a/b) = b/a.
Describe g–1.
g–1(a/b) = b/a.
Which of the four basic properties does g–1 have?

8. Example: Inverse Relation
Let g : Q  Q by h(a/b) = (a/b)2.
Describe h–1.
Which of the four basic properties does h–1 have?

9. Inverse Relations and the Basic Properties
A relation R has the first basic property if and only if R–1 has the third basic property.
x  A,  at least one y  B, (x, y)  R.
y  B,  at least one x  A, (x, y)  R.
A relation R has the second basic property if and only if R–1 has the fourth basic property.
x  A,  at most one y  B, (x, y)  R.
y  B,  at most one x  A, (x, y)  R.

10. Inverse Functions
The inverse of a function is, in general, a relation, but not a function.
Theorem: The inverse of a function is itself a function if and only if the function is a one-to-one correspondence.
Corollary: If f is a one-to-one correspondence, then f –1 is a one-to-one correspondence.

11. Example: Inverse Functions
Let A = R – {1/3}.
Let B = R – {2/3}.
Define f : A  B by f(x) = 2x/(3x – 1).
Find f –1.
Let y = 2x/(3x – 1).
Swap x and y: x = 2y/(3y – 1).
Solve for y: y = x/(3x – 2).
Therefore, f –1(x) = x/(3x – 2).

12. Q and Z Theorem: There is a one-to-one correspondence from Z to Q.
Proof:
Consider only rationals in reduced form.
Arrange the positive rationals in order
First by the sum of numerator and denominator.
Then, within groups, by numerator.

13. Q and Z The sequence is 1/1, 1/2, 2/1, 1/3, 3/1, 1/4, 2/3, 3/2, 4/1, …
The first group: 1/1
The second group: 1/2, 2/1
The third group: 1/3, 3/1
The fourth group: 1/4, 2/3, 3/2, 4/1
Etc.
The sequence is 1/1, 1/2, 2/1, 1/3, 3/1, 1/4, 2/3, 3/2, 4/1, …

14. Q and Z Let f : Z to Q be the function that
Maps the positive integer n to the nth rational in this list.
Maps the negative integer -n to the negative of the rational that n maps to.
Maps 0 to 0.
This is a one-to-one correspondence.

15. Q and Z
What is f(20)?
What is f –1(4/5)?