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:

one-one (or 1-1) or one-to-one for injective

one-one mapping or one-to-one mapping for injection.

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

• Equivalence of Definitions of Injection

• Results about injections can be found here.

Sources

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