Injection

An injection, or "one-to-one function," is a function that takes distinct values on distinct inputs. Equivalently, an injection is a function for which every value in the range is the image of exactly one value in the domain.

Alternative definition: A function $f:A\to B$ is an injection if for all $x,y\in A$ , if $f(x)=f(y)$ then $x=y$ .

The binary relation $|X|\leq|Y|$ iff there is an injection $f:X\rightarrow Y$ forms a partial order on the class of cardinals: $X\leq X$ , $X\leq Y$ and $Y\leq X$ implies $|X|=|Y|$ by the Cantor-Schroeder-Bernstein theorem, and $|X|\leq|Y|$ and $|Y|\leq|Z|$ implies $|X|\leq|Z|$ because the composition of injections is again an injection.

Examples

Linear functions are injections: $f:\mathbb R \to \mathbb R$ , $f(x)= ax+b$ , $a\neq 0$ . The domain choosing is also important. For example, while $f:\mathbb R \to \mathbb R$ , $f(x)=x^2$ is not an injection ( $f(-1)=f(1)=1$ ), the function $g:[0,\infty)\to\mathbb R$ , $g(x)=x^2$ , is an injection.

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Injection

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