1. One-to-One and Onto, Inverse Functions
Lecture 36
Section 7.2
Mon, Apr 2, 2007

2. Four Important Properties
Let R be a relation from A to B.
R may have any of the following four properties.
x  A,  at least one y  B, (x, y)  R.
x  A,  at most one y  B, (x, y)  R.
y  B,  at least one x  A, (x, y)  R.
y  B,  at most one x  A, (x, y)  R.

3. One-to-one and Onto R is onto if R is one-to-one if
y  B,  at least one x  A, (x, y)  R.
R is one-to-one if
y  B,  at most one x  A, (x, y)  R.

4. Combinations of Properties
R is a function if
x  A,  at exactly one y  B, (x, y)  R.
R is one-to-one and onto if
y  B,  at exactly one x  A, (x, y)  R.

5. Examples Consider the following “functions.” f : R  R by f(x) = 2x.
g : R  R by g(x) = 1/x.
h : R  R by h(x) = x2.
k : R  R by m(x) = x.
m : Q  Q by k(a/b) = a.

6. Examples Which of them are functions?
Prove that g : R*  R is one-to-one, but not onto.
Is it one-to-one and onto from R* to R*?
Prove that h is neither one-to-one nor onto.
What about k?
What about m?

7. 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.

8. 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.

9. Example: One-to-one Correspondence
Are any of the following functions one-to-one correspondences?
f : R  R by f(x) = 2x.
g : R*  R* by g(x) = 1/x.
h : R  R by h(x) = x2.
m : R  R by k(x) = x.

10. Inverse Relations Let R be a relation from A to B.
The inverse relation 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.
If a function f : A  B is a one-to-one correspondence, then it has an inverse function f -1 : B  A such that if f(x) = y, then f -1(y) = x.

11. Example: Inverse Relation
Let f : R  R by f(x) = 2x.
Describe f –1.

12. Example: Inverse Relation
Let g : R*  R* by g(x) = 1/x.
Describe g–1.

13. Example: Inverse Relation
Let k : R  R by k(x) = x.
Describe k–1.

14. 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).

15. Example: Inverse Relation
Let A = R and B = R.
Let j : A  B by j(x) = (3x – 1)/(x + 1).
Find j -1.
What values must be deleted from A and B to make j a one-to-one correspondence?
Verify that the modified j is one-to-one and onto.

16. 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.

17. Inverse Relations and the Basic Properties
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.

18. Inverse Functions
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.
The inverse of a function is, in general, a relation, but not a function.

19. 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.

20. 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, …

21. Q and Z Let f : Z  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.

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