One-to-One and Onto, Inverse Functions Lecture 31 Section 7.2 Thu, Apr 6, 2006

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.

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.

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.

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.

Examples Which of the five examples are functions? Prove that g is one-to-one, but not onto. Is it one-to-one and onto from Q – {0} to Q – {0}? Prove that h is neither one-to-one nor onto. What about k? What about m?

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.

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.

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. Is g a one-to-one correspondence from R – {0} to R – {0}?

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.

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

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

Example: Inverse Relation Let h : R  R by h(x) = x2. Describe h–1. Which of the four basic properties does h–1 have?

Example: Inverse Relation Let k : R  R by k(x) = x. Describe k–1. Which of the four basic properties does k–1 have?

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.

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.

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.

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).

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.

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, …

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.

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