Equivalence of Definitions of Bijection/Definition 3 iff Definition 5
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Theorem
The following definitions of the concept of Bijection are equivalent:
Definition 3
A mapping $f: S \to T$ is a bijection if and only if:
Definition 5
A relation $f \subseteq S \times T$ is a bijection if and only if:
- $(1): \quad$ for each $x \in S$ there exists one and only one $y \in T$ such that $\tuple {x, y} \in f$
- $(2): \quad$ for each $y \in T$ there exists one and only one $x \in S$ such that $\tuple {x, y} \in f$.
Proof
Necessary Condition
Let $f: S \to T$ be a mapping such that $f^{-1}: T \to S$ is also a mapping.
Then as $f$ is a mapping:
- for all $x \in S$ there exists a unique $y \in T$ such that $\tuple {x, y} \in f$.
Similarly, as $f^{-1}$ is also a mapping:
- for all $y \in T$ there exists a unique $x \in S$ such that $\tuple {y, x} \in f^{-1}$.
$\Box$
Sufficient Condition
Let $f \subseteq S \times T$ be a relation such that:
- $(1): \quad$ for each $x \in S$ there exists one and only one $y \in T$ such that $\tuple {x, y} \in f$
- $(2): \quad$ for each $y \in T$ there exists one and only one $x \in S$ such that $\tuple {x, y} \in f$.
Then by definition:
$\blacksquare$