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Symbolic Condition for a One-One Function

publish date: 2026/05/23 18:47:47.005370 UTC

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A function \(f: A \to B\) is one-one (injective) if and only if:

\(f(a_1) \ne f(a_2) \Rightarrow a_1 = a_2\)

\(f(a_1) = f(a_2) \Rightarrow a_1 = a_2\) for every \(a_1, a_2 \in A\)

\(a_1 = a_2 \Rightarrow f(a_1) \ne f(a_2)\)

Every element of B has exactly two pre-images in A

Correct Answer

\(f(a_1) = f(a_2) \Rightarrow a_1 = a_2\) for every \(a_1, a_2 \in A\)

Explanation

The formal definition of injectivity: for every \(a_1, a_2 \in A\), \(f(a_1) = f(a_2) \Rightarrow a_1 = a_2\). Equivalently (the contrapositive), \(a_1 \ne a_2 \Rightarrow f(a_1) \ne f(a_2)\). Both forms say the same thing: equal outputs imply equal inputs, i.e., no two different inputs share an output.

Reference

Introduction to Differential Calculus (Systematic Studies with Engineering Applications for Beginners) - 2012

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