The function is also surjective, because the codomain coincides with the range. A homomorphism between algebraic structures is a function that is compatible with the operations of the structures. In a metric space it is an isometry. It is injective (any pair of distinct elements of the domain is mapped to distinct images in the codomain). But having an inverse function requires the function to be bijective. a ≠ b ⇒ f(a) ≠ f(b) for all a, b ∈ A f(a) […] We also say that \(f\) is a one-to-one correspondence. Then your question reduces to 'is a surjective function bijective?' Surjective is where there are more x values than y values and some y values have two x values. A function is injective if no two inputs have the same output. Bijective is where there is one x value for every y value. And in any topological space, the identity function is always a continuous function. bijective if f is both injective and surjective. So, let’s suppose that f(a) = f(b). However, sometimes papers speaks about inverses of injective functions that are not necessarily surjective on the natural domain. Since the identity transformation is both injective and surjective, we can say that it is a bijective function. Or let the injective function be the identity function. A function \(f : A \to B\) is said to be bijective (or one-to-one and onto) if it is both injective and surjective. The range of a function is all actual output values. In other words, if you know that $\log$ exists, you know that $\exp$ is bijective. $\endgroup$ – Aloizio Macedo ♦ May 16 '15 at 4:04 $\begingroup$ Injective is where there are more x values than y values and not every y value has an x value but every x value has one y value. Surjective Injective Bijective: References Below is a visual description of Definition 12.4. It is also not surjective, because there is no preimage for the element \(3 \in B.\) The relation is a function. The term surjective and the related terms injective and bijective were introduced by Nicolas Bourbaki, a group of mainly French 20th-century mathematicians who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. Theorem 4.2.5. Injective, Surjective and Bijective One-one function (Injection) A function f : A B is said to be a one-one function or an injection, if different elements of A have different images in B. 1. No, suppose the domain of the injective function is greater than one, and the surjective function has a singleton set as a codomain. Accelerated Geometry NOTES 5.1 Injective, Surjective, & Bijective Functions Functions A function relates each element of a set with exactly one element of another set. The codomain of a function is all possible output values. Let f: A → B. Thus, f : A B is one-one. Is it injective? 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