Understanding Injective Functions
Injective functions, or one-to-one functions, have a unique property that distinguishes them from other types of functions. Let's explore the key aspects to understand why option 'A' is the correct answer.
Definition of Injective Function
- A function f: X → Y is considered injective if every element in the domain X maps to a unique element in the codomain Y.
Analyzing Option A
- Option 'A' states: "f(x1) = f(x2) ⇒ x1 = x2, for all x1, x2 ∈ X."
- This means if two inputs produce the same output, then those inputs must be identical. This is a fundamental characteristic of injective functions.
Why Other Options Are Incorrect
- Option B: "x1 ≠ x2 ⇒ f(x1) = f(x2)" suggests that distinct inputs lead to the same output, which contradicts the definition of injectivity.
- Option C: "f(X) = Y" implies that the function is surjective (onto), but injectivity does not guarantee that every element in Y is mapped.
- Option D: "X = Y" states that the sets must be equal, which is not a requirement for a function to be injective. Injectivity is about the relationship between the elements, not the sizes of the sets.
Conclusion
- Therefore, the essence of an injective function lies in the relationship between inputs and outputs, clearly captured by option 'A'. If two outputs are equal, the inputs must also be equal, affirming the one-to-one nature of the function.