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## Definition: Injective Function

A function $$f:A\mapsto B$$ is called injective, (or injection, or embedding, or, less formally, one-to-one), if for all $$a_1,a_2\in A$$ with $f(a_1)=f(a_2)$ it follows that $a_1=a_2.$ This corresponds to the left-unique property, in addition to the defining properties of a function. By a contraposition argument the following definition is equivalent: For all $$a_1,a_2\in A$$, it follows from $a_1\neq a_2$ that $f(a_1)\neq f(a_2).$

### Notes

• An injective function allows for every $b\in B$ at most an $a\in A$ such that $f(a)=b.$
• In other words, either there is exactly one or no such element $a\in A.$

### Bibliography (further reading)

 Kane, Jonathan: “Writing Proofs in Analysis”, Springer, 2016

 Knauer Ulrich: “Diskrete Strukturen – kurz gefasst”, Spektrum Akademischer Verlag, 2001

 Reinhardt F., Soeder H.: “dtv-Atlas zur Mathematik”, Deutsche Taschenbuch Verlag, 1994, 10