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One-to-One and Onto Relations: Lecture on Properties and Inverses

Explore the crucial concepts of one-to-one, onto relations, and inverse functions with examples and proofs of basic properties. Dive into the significance of one-to-one correspondences.

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One-to-One and Onto Relations: Lecture on Properties and Inverses

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

  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 : 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. • k : QQ by k(a/b) = a. • m : QQ by m(a/b) = a/b.

  6. Examples • Which of the five examples are functions? • Prove that g is one-to-one, but not onto. • Prove that h is neither one-to-one nor onto. • 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 : 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}?

  10. 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.

  11. 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?

  12. 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?

  13. 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?

  14. 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.

  15. 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.

  16. 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.

  17. 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).

  18. 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.

  19. 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, …

  20. 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.

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

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