1 / 22

One-to-One and Onto, Inverse Functions

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

tait
Download Presentation

One-to-One and Onto, Inverse Functions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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 • 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-onecorrespondence 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 functionf-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 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)?

More Related