Math Problem Statement

Prove or disprove: Every function f : ℝ → ℝ that is a bijection is either strictly increasing or strictly decreasing. Prove or disprove: Every function f : ℝ → ℝ that is either strictly increasing or strictly decreasing is a bijection.

Solution

Let's break down and address the two parts of the question in the image:

(a) Prove or Disprove:

Statement: Every function f:R→Rf : \mathbb{R} \to \mathbb{R} that is a bijection is either strictly increasing or strictly decreasing.

Solution:

A bijection means the function is both injective (one-to-one) and surjective (onto). The claim suggests that any bijection from R\mathbb{R} to R\mathbb{R} must be either strictly increasing or strictly decreasing. We will consider whether this is necessarily true for all bijections.

  • If a function is strictly increasing or strictly decreasing, then it must be injective because two different values in the domain map to different values in the codomain.
  • Additionally, if the function is surjective, it must map every value of R\mathbb{R} onto R\mathbb{R}, meaning no value in the codomain is missed.

While it's true that any strictly increasing or strictly decreasing function would satisfy the conditions for a bijection, there exist functions that are bijections but are neither strictly increasing nor strictly decreasing.

Counterexample:

Consider the piecewise function f(x)f(x) defined as:

\begin{cases} x, & \text{if } x \geq 0 \\ x - 2, & \text{if } x < 0 \end{cases}$$ - This function is **bijective** because every point in $$\mathbb{R}$$ is mapped uniquely, and the function is onto. - However, it is neither strictly increasing nor strictly decreasing because at $$x = 0$$, the function has a discontinuity. Thus, the statement is **false**. Not every bijection is strictly increasing or strictly decreasing. ### (b) Prove or Disprove: **Statement:** Every function $$f : \mathbb{R} \to \mathbb{R}$$ that is either strictly increasing or strictly decreasing is a bijection. #### Solution: Let’s analyze this step by step. - A function that is strictly increasing or strictly decreasing will be injective because, by definition, no two distinct points $$x_1$$ and $$x_2$$ will have the same function value. In other words, $$f(x_1) \neq f(x_2)$$ if $$x_1 \neq x_2$$. - To check surjectivity (onto), note that for a function $$f : \mathbb{R} \to \mathbb{R}$$, since $$f(x)$$ either strictly increases or strictly decreases without any bounds, it must map every real number in the codomain $$\mathbb{R}$$. Thus, it covers the entire real line. Therefore, if a function is strictly increasing or strictly decreasing, it will indeed be a bijection. Hence, the statement is **true**. --- Do you want more details on any part? Feel free to ask! Here are some related questions: 1. What other examples can be used to prove a bijection that isn't strictly monotonic? 2. How can we define bijections for functions that are not continuous? 3. How does the concept of monotonicity relate to differentiability? 4. Can a piecewise function be a bijection without being monotonic? 5. What role do inverses of functions play in the study of bijections? **Tip:** A strictly increasing or decreasing function has a well-defined inverse, which can simplify analyzing bijections.

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Math Problem Analysis

Mathematical Concepts

Functions
Bijections
Monotonicity
Real Analysis

Formulas

-

Theorems

Definition of a Bijection
Monotonicity Theorem

Suitable Grade Level

Undergraduate (Math Majors or Real Analysis Course)

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