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Is a One-One Function Always Onto?
publish date: 2026/05/23 18:47:52.337242 UTC
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True or False: A one-one (injective) function \(f: A \to B\) is always also onto (surjective).
Correct Answer
False
Explanation
False. An injective function need not be surjective. In the first diagram of this quiz, \(a, b, c\) map injectively to \(1, 2, 3\) — but element \(4\) in the codomain has no pre-image, so the function is not onto. Only when a function is both injective and surjective is it bijective.
Reference
Introduction to Differential Calculus (Systematic Studies with Engineering Applications for Beginners) - 2012