Surjective is where there are more x values than y values and some y values have two x values. Then your question reduces to 'is a surjective function bijective?' 1. $\endgroup$ – Aloizio Macedo ♦ May 16 '15 at 4:04 The point is that the authors implicitly uses the fact that every function is surjective on it's image . $\endgroup$ – Wyatt Stone Sep 7 '17 at 1:33 No, suppose the domain of the injective function is greater than one, and the surjective function has a singleton set as a codomain. So, let’s suppose that f(a) = f(b). 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. Below is a visual description of Definition 12.4. 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. In essence, injective means that unequal elements in A always get sent to unequal elements in B. Surjective means that every element of B has an arrow pointing … a ≠ b ⇒ f(a) ≠ f(b) for all a, b ∈ A f(a) […] Theorem 4.2.5. The domain of a function is all possible input values. Bijective is where there is one x value for every y value. We also say that \(f\) is a one-to-one correspondence. It is also not surjective, because there is no preimage for the element \(3 \in B.\) The relation is a function. Surjective Injective Bijective: References It is injective (any pair of distinct elements of the domain is mapped to distinct images in the codomain). The function is also surjective, because the codomain coincides with the range. However, sometimes papers speaks about inverses of injective functions that are not necessarily surjective on the natural domain. The codomain of a function is all possible output values. 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. A homomorphism between algebraic structures is a function that is compatible with the operations of the structures. A function \(f : A \to B\) is said to be bijective (or one-to-one and onto) if it is both injective and surjective. But having an inverse function requires the function to be bijective. Then 2a = 2b. $\begingroup$ 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. Since the identity transformation is both injective and surjective, we can say that it is a bijective function. In a metric space it is an isometry. bijective if f is both injective and surjective. 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