7.2 One-to-One, Onto, and Inverse Functions

Definitions

• Let F be a function from a set X to a set Y. F is one-to-one (or injective) if and only if for all elements x₁ and x₂ in X,
  • if F(x₁) = F(x₂), then x₁ = x₂
  • (or equivalently) if x₁ ≠ x₂ then F(x₁) ≠ F(x₂)
  • Symbolically: F: X → Y is one-to-one ↔ ∀ x₁, x₂ ∈ X, if F(x₁) = F(x₂) then x₁ = x₂
• Let F be a function from a set X to a set Y. F is onto (or surjective) if and only if given any element y in Y, it is possible to find an element x in X with the property that y = F(x).
  • Symbolically: F: X → Y is onto ↔ ∀ y ∈ Y, ∃ x ∈ X such that F(x) = y
• A one-to-one correspondence (or bijection) from a set X to a set Y is a function F: X → Y that is both one-to-one and onto

Proof Techniques

Suppose f is a function defined on an infinite set X

• To prove f is one-to-one, (1) suppose x₁ and x₂ are elements of X such that f(x₁) = f(x₂), and then (2) show that x₁ = x₂
• To prove that f is not one-to-one, find x₁ and x₂ in X so that f(x₁) = f(x₂) but x₁ ≠ x₂.
• To prove f is onto, (1) suppose that y is any element of Y and then (2) show that there is an element x in X with F(x) = y
• To prove that f is not onto, find an element y of Y such that y ≠ F(x) for any x in X

One-to-One Data Structure Application

• Hash Function: A function defined from a larger, possibly infinite, set of data to a smaller fixed-size set of integers.
• Note: Hash functions are not guaranteed to be one-to-one so a collision resolution strategy is needed.

Theorem 7.2.2

Suppose F: X → Y is a one-to-one correspondence; in other words, suppose F is one-to-one and onto. Then there is a function F^-1: Y → X that is defined as follows: Given any element y in Y,

• F^-1(y) = that unique element x in X such that F(x) equals y
• Equivalently, F^-1(y) = x ↔ y = F(x)

The function F^-1 is called the inverse function for F.

Theorem 7.2.3

If X and Y are sets and F: X → Y is one-to-one and onto, then F^-1:Y → X is also one-to-one and onto.

In Class Exercises

• 6a. Let X = {1, 5, 9}, Y = {3, 4, 7}, f(1) = 4, f(5) = 7 and f(9) = 4. Is f one-to-one? Is f onto? Explain your answers.
• 10a. Define f: Z → Z by the rule f(n) = 2n for every integer n. Is f one-to-one? Is f onto? Prove or give a counterexample.
• 26. Define S: Z⁺ → Z⁺ by the rule: For each integer n, S(n) = the sum of the positive divisors of n. Is S one-to-one? Is S onto? Prove or give a counterexample.
• 44. Is function f in 6a a one-to-one correspondence? If so, find f^-1. If not modify it to be a one-to-one correspondence and then find f^-1

A Few Set Applications (Chapter 6)

• Data Structures (CSCI 132, CSCI 232, CSCI 432)
• Databases (CSCI 440)
• Machine Learning (CSCI 446)
• Formal Languages (CSCI 338)