Definition:Bijection/Definition 1

From ProofWiki
Jump to navigation Jump to search

Definition

A mapping $f: S \to T$ is a bijection if and only if both:

$(1): \quad f$ is an injection

and:

$(2): \quad f$ is a surjection.


That is, if and only if $f$ is a relation which is:

$(1): \quad$ left-total
$(2): \quad$ right-total
$(3): \quad$ functional (many-to-one)
$(4): \quad$ injective (one-to-many).


Also known as

The terms

biunique correspondence
bijective correspondence

are sometimes seen for bijection.

Authors who prefer to limit the jargon of mathematics tend to use the term one-one and onto mapping for bijection.

If a bijection exists between two sets $S$ and $T$, then $S$ and $T$ are said to be in one-to-one correspondence.

Some sources refer to exact correspondence to mean exactly this.


Occasionally you will see the term set isomorphism, but the term isomorphism is usually reserved for mathematical structures of greater complexity than a set.

Some authors, developing the concept of inverse mapping independently from that of the bijection, call such a mapping invertible.


The symbol $f: S \leftrightarrow T$ is sometimes seen to denote that $f$ is a bijection from $S$ to $T$.

Also seen sometimes is the notation $f: S \cong T$ or $S \stackrel f \cong T$ but this is cumbersome and the symbol $\cong$ already has several uses.


In the context of class theory, a bijection is often seen referred to as a class bijection.


Technical Note

The $\LaTeX$ code for \(f: S \leftrightarrow T\) is f: S \leftrightarrow T .

The $\LaTeX$ code for \(f: S \cong T\) is f: S \cong T .

The $\LaTeX$ code for \(S \stackrel f \cong T\) is S \stackrel f \cong T .


Also see

  • Results about bijections can be found here.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Bijection/Definition_1&oldid=707125"