How To Prove One To One

How to Prove a Function is One-to-One (Injective)

One-to-one functions, also known as injective functions, are a fundamental concept in mathematics, particularly in areas like calculus, linear algebra, and abstract algebra. Understanding how to prove a function is one-to-one is crucial for various applications, including cryptography, coding theory, and the study of mathematical structures. This comprehensive guide will walk you through various methods and strategies for proving injectivity, equipping you with the tools to tackle a wide range of function types.

Understanding One-to-One Functions

Before diving into the methods of proof, let's solidify our understanding of what a one-to-one function actually is. A function f: A → B is one-to-one (or injective) if every element in the codomain B is mapped to by at most one element in the domain A. In simpler terms, this means that no two distinct elements in the domain map to the same element in the codomain. Formally, we can state this as:

For all x₁, x₂ ∈ A, if f(x₁) = f(x₂), then x₁ = x₂.

This definition forms the basis of most proofs of injectivity. We often employ the contrapositive of this statement, which is logically equivalent:

For all x₁, x₂ ∈ A, if x₁ ≠ x₂, then f(x₁) ≠ f(x₂).

This contrapositive states that if two distinct elements in the domain are different, then their images under the function must also be different. Both the original statement and its contrapositive are equally valid for proving injectivity.

Methods for Proving Injectivity

Several methods can be used to prove a function is one-to-one. The choice of method often depends on the nature of the function itself.

1. Direct Proof Using the Definition

This is the most straightforward method. We directly apply the definition of a one-to-one function. We start by assuming that f(x₁) = f(x₂), and then we use algebraic manipulation or logical reasoning to show that this implies x₁ = x₂.

Example: Prove that the function f(x) = 3x + 2 is one-to-one.

Proof: Let x₁, x₂ ∈ ℝ such that f(x₁) = f(x₂). This means:

3x₁ + 2 = 3x₂ + 2

Subtracting 2 from both sides gives:

3x₁ = 3x₂

Dividing both sides by 3 gives:

x₁ = x₂

Therefore, f(x) = 3x + 2 is one-to-one.

2. Proof Using the Contrapositive

As mentioned earlier, using the contrapositive can sometimes simplify the proof. We assume x₁ ≠ x₂ and show that f(x₁) ≠ f(x₂).

Example: Prove that the function f(x) = x² is not one-to-one for x ∈ ℝ.

Proof: Let's consider x₁ = 2 and x₂ = -2. Clearly, x₁ ≠ x₂. However, f(x₁) = 2² = 4 and f(x₂) = (-2)² = 4. Since f(x₁) = f(x₂), but x₁ ≠ x₂, the function is not one-to-one. This is a counterexample which disproves injectivity.

3. Graphical Method

For functions with simple graphical representations, we can use the horizontal line test. If any horizontal line intersects the graph of the function at more than one point, the function is not one-to-one. If every horizontal line intersects the graph at most once, the function is one-to-one. This method is intuitive but only works for functions that can be easily graphed.

4. Calculus Techniques (for differentiable functions)

If the function is differentiable, we can use calculus techniques to prove injectivity. Specifically, if the function is strictly monotonic (either strictly increasing or strictly decreasing) on its domain, then it is one-to-one. We can determine monotonicity by analyzing the derivative:

• Strictly increasing: If f′(x) > 0 for all x in the domain, then f(x) is strictly increasing and therefore one-to-one.
• Strictly decreasing: If f′(x) < 0 for all x in the domain, then f(x) is strictly decreasing and therefore one-to-one.

Example: Prove that f(x) = eˣ is one-to-one.

Proof: The derivative of f(x) = eˣ is f′(x) = eˣ. Since eˣ > 0 for all x ∈ ℝ, f(x) is strictly increasing and therefore one-to-one.

5. Proof by Contradiction

This method involves assuming the negation of the statement we want to prove and then showing that this assumption leads to a contradiction.

Example: Prove that f(x) = 1/x is one-to-one for x ∈ (0, ∞).

Proof: Assume, for the sake of contradiction, that f(x) is not one-to-one on (0, ∞). Then there exist distinct x₁, x₂ ∈ (0, ∞) such that f(x₁) = f(x₂). This implies 1/x₁ = 1/x₂, which in turn implies x₁ = x₂. This contradicts our assumption that x₁ and x₂ are distinct. Therefore, f(x) = 1/x is one-to-one on (0, ∞).

6. Using Properties of Injections

Certain properties of injections can simplify proofs. For instance, the composition of two injections is always an injection.

Dealing with Different Function Types

The approach to proving injectivity will vary based on the type of function:

Polynomial Functions:

For polynomial functions, direct proof or proof by contradiction are commonly used. Consider carefully the degree of the polynomial and the domain. A polynomial of degree greater than one is rarely one-to-one across its entire domain.

Trigonometric Functions:

Trigonometric functions are rarely one-to-one across their entire domains. You usually need to restrict the domain to a specific interval to demonstrate injectivity (e.g., sin(x) is injective on [-π/2, π/2]). Careful consideration of the range and domain is paramount.

Exponential and Logarithmic Functions:

Exponential functions (like eˣ) are generally one-to-one across their entire domain. Their monotonicity makes proving injectivity straightforward using the derivative. Similarly, logarithmic functions are also often one-to-one on their respective domains.

Piecewise Functions:

For piecewise functions, you need to prove injectivity on each piece of the domain separately. This often involves separate analyses for each subdomain where the function's definition changes.

Common Pitfalls to Avoid

• Confusing one-to-one with onto (surjective): A one-to-one function maps each element in the domain to a unique element in the codomain. An onto function maps every element in the codomain to at least one element in the domain. These are distinct concepts.
• Incorrectly applying the horizontal line test: The horizontal line test is a visual tool and is not a rigorous proof.
• Failing to consider the domain and codomain: The domain and codomain are crucial in determining if a function is one-to-one. A function may be one-to-one on a restricted domain but not on a larger domain.
• Ignoring counterexamples: If you can find even one example that violates the definition of one-to-one, you have disproven injectivity.

Conclusion: Mastering the Art of Proving Injectivity

Proving a function is one-to-one is a fundamental skill in mathematics with applications across numerous fields. By understanding the different methods presented in this guide – direct proof, contrapositive, graphical method, calculus techniques, proof by contradiction, and utilizing the properties of injections – you’ll be well-equipped to tackle a wide variety of functions and domains. Remember to always carefully consider the specific nature of the function, its domain and codomain, and choose the most appropriate proof technique. Practice is key to mastering this essential mathematical concept. The more examples you work through, the better you’ll become at identifying the most efficient and elegant approach for each problem.