Equivalence of Definitions of Bijection/Definition 1 iff Definition 4
Theorem
The following definitions of the concept of Bijection are equivalent:
Definition 1
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.
Definition 4
A mapping $f \subseteq S \times T$ is a bijection if and only if:
- for each $y \in T$ there exists one and only one $x \in S$ such that $\tuple {x, y} \in f$.
Proof
Let $f: S \to T$ be a bijection by definition 1.
Then by definition:
- $f$ is an injection
- $f$ is a surjection
By definition of injection:
By definition of surjection:
So:
- for each $y \in T$ there exists one and only one $x \in S$ such that $\tuple {x, y} \in f$.
Thus $f$ is a bijection by definition 4.
$\Box$
Let $f: S \to T$ be a bijection by definition 4.
Then by definition:
- for each $y \in T$ there exists one and only one $x \in S$ such that $\tuple {x, y} \in f$.
But:
defines an injection
and:
defines a surjection.
From Injection iff Left Inverse, $f$ is an injection if and only if $f$ has a left inverse mapping.
From Surjection iff Right Inverse, $f$ is a surjection if and only if $f$ has a right inverse mapping.
Putting these together, it follows that:
- $f$ is an injection
- $f$ is a surjection
Thus $f$ is a bijection by definition 1.
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 22$: Injections; bijections; inverse of a bijection