Properties of Injective Function

There are various properties of Injecive functions, some of those are listed as follows:

• For every input element in the domain of the function, there is a unique output element in the codomain.
• Injective functions are often monotonic i.e., function is either strictly increases or strictly decreases as you move along the real number line.
• An injective function does not have any critical points (i.e., points where the derivative is zero or undefined) within its domain.
• An injective function that is also surjective (onto) is called a bijective function.

Some more properties of Injective function include:

• The composition of two injective functions is also an injective function.
  • If f: A → B and g: B → C are both injective functions, then their composition g(f(x)) is also injective.
• Two sets A and B have the same cardinality if and only if there exists an injective function from A to B and an injective function from B to A.
• If you have a function f: A → B and a subset A’ of A, you can restrict the domain of f to A’ to create a new function. This restricted function is still injective if f is injective on A’.

Injective Functions, also called one-to-one functions are a fundamental concept in mathematics because they establish a unique correspondence between elements of their domain and codomain, ensuring that no two distinct elements in the domain map to the same element in the codomain.

In the world of injective functions, consider two sets, Set A and Set B. The unique characteristic of injective functions is that every element in Set A corresponds to a distinct element in Set B and there is no sharing or repetition. This article provides a well-rounded description of the concept of “Injective Function”, including definitions, examples, properties, and many more.