A homomorphism between algebraic structures is a function that is compatible with the operations of the structures. Bijective is where there is one x value for every y value. A non-injective non-surjective function (also not a bijection) . $\begingroup$ 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. Theorem 4.2.5. A function is injective if no two inputs have the same output. Or let the injective function be the identity function. The domain of a function is all possible input values. 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. Below is a visual description of Definition 12.4. The function is also surjective, because the codomain coincides with the range. Thus, f : A B is one-one. Surjective Injective Bijective: References So, let’s suppose that f(a) = f(b). Let f: A → B. bijective if f is both injective and surjective. The range of a function is all actual output values. A function $$f : A \to B$$ is said to be bijective (or one-to-one and onto) if it is both injective and surjective. Then 2a = 2b. Dividing both sides by 2 gives us a = b. 1. In a metric space it is an isometry. We also say that $$f$$ is a one-to-one correspondence. But having an inverse function requires the function to be 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 … Then your question reduces to 'is a surjective function bijective?' The codomain of a function is all possible output values. $\endgroup$ – Wyatt Stone Sep 7 '17 at 1:33 In other words, if you know that $\log$ exists, you know that $\exp$ is bijective. No, suppose the domain of the injective function is greater than one, and the surjective function has a singleton set as a codomain. a ≠ b ⇒ f(a) ≠ f(b) for all a, b ∈ A f(a) […] The point is that the authors implicitly uses the fact that every function is surjective on it's image . Surjective is where there are more x values than y values and some y values have two x values. $\endgroup$ – Aloizio Macedo ♦ May 16 '15 at 4:04 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. Is it injective? However, sometimes papers speaks about inverses of injective functions that are not necessarily surjective on the natural domain. 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