Definition:Bijection/Definition 1

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

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


$(2): \quad f$ 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.

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