Injective, Surjective, Bijective

math [Bijection], [Surjection], [Injection]

Function $f(x):A\to B$ is called surjective, when

$$(\forall y\in B)(\exists x\in A)(f(x)=y)$$

Surjectivity means that “every image $y$ has at least one pre-image $x$“.

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Function $f(x):A\to B$ is called injective, when

$$(\forall x,y\in A)(f(x)=f(y)\Rightarrow x=y)$$

Injectivity means that “every image $y$ has at most one pre-image $x$“.

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Function $f(x):A\to B$ is called bijective, when it is both surjective and injective.

Which is usefull definition for proofs. It is sometimes defined as each image y belongs exactly to one element x. It also implies $|A|=|B|$ for finite sets. In case of sets with infinite cardinality, existence of bijection between those sets is what defines their equal cardinality.

Examples

1. Identity function $f(x)=x$ is bijection.
2. Quadratic function $f(x)=x^2$ is neither injective, nor surjective when $f:\mathbb{R}\to \mathbb{R}$. However, it is bijective when $f:(0,\infty )\to (0,\infty )$. That is because for every $y\in \mathbb{R}\setminus \{0\}$, there exists $x=\pm \sqrt{y}$.