# Definition:Injection/Definition 2

## Definition

An **injection** is a relation which is both one-to-one and left-total.

Thus, a relation $f$ is an **injection** if and only if:

- $(1): \quad \forall x \in \Dom f: \tuple {x, y_1} \in f \land \tuple {x, y_2} \in f \implies y_1 = y_2$
- $(2): \quad \forall y \in \Img f: \tuple {x_1, y} \in f \land \tuple {x_2, y} \in f \implies x_1 = x_2$
- $(3): \quad \forall s \in S: \exists t \in T: \tuple {s, t} \in \RR$

## Also known as

Authors who prefer to limit the jargon of mathematics tend to use the term:

However, because of the possible confusion with the term one-to-one correspondence, it is standard on $\mathsf{Pr} \infty \mathsf{fWiki}$ for the technical term **injection** to be used instead.

E.M. Patterson's idiosyncratic *Topology, 2nd ed.* of $1959$ refers to such a mapping as **biuniform**.

This is confusing, because a casual reader may conflate this with the definition of a **bijection**, which in that text is not explicitly defined at all.

An **injective mapping** is sometimes written:

- $f: S \rightarrowtail T$ or $f: S \hookrightarrow T$

In the context of class theory, an **injection** is often seen referred to as a **class injection**.

## Also see

- Results about
**injections**can be found**here**.

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 6.4$