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:
| \(\text {(1)}: \quad\) | \(\ds \forall x \in \Dom f: \, \) | \(\ds \tuple {x, y_1} \in f \land \tuple {x, y_2} \in f\) | \(\implies\) | \(\ds y_1 = y_2\) | ||||||||||
| \(\text {(2)}: \quad\) | \(\ds y \in \Img f: \, \) | \(\ds \tuple {x_1, y} \in f \land \tuple {x_2, y} \in f\) | \(\implies\) | \(\ds x_1 = x_2\) | ||||||||||
| \(\text {(3)}: \quad\) | \(\ds \forall s \in S: \exists t \in T: \, \) | \(\ds \tuple {s, t}\) | \(\in\) | \(\ds \RR\) |
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 or one-to-one map 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
- Results about injections can be found here.
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 6.4$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): injection
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): injection