What is the meaning of injective?

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The concept of an injective function embodies a fundamental property in mathematics, where for any given function f, if f(x) equals f(y), it necessarily implies that x equals y. This characteristic distinguishes injective functions from others, ensuring a unique mapping from the domain to the codomain.

Conversely, the definition of a partial function specifies a relationship where the implication x=y⇒f(x)=f(y) holds. This means that when two elements are equal, their function values ​​under f must also be equal, but it does not necessarily enforce the reverse implication, differentiating it from an injective function.

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By combining the properties of both injective and partial functions, we arrive at the notion of an injective partial function. Here, the defining characteristic is that f(x)=f(y) if and only if x=y. This condition ensures both the uniqueness of mapping and the applicability of the function to a subset of its domain.