Proper Subset of Finite Set No Bijection

Theorem

A finite set can not be in one-to-one correspondence with one of its proper subsets.

That is, a finite set is not Dedekind-infinite.

Proof 1

Let $S$ be a finite set, and let $T$ be a proper subset of $S$.

Let $f : T \to S$ be an injection.

By Cardinality of Image of Injection and Cardinality of Subset of Finite Set, $\left\vert{\operatorname{im} \left({f}\right)}\right\vert = \left\vert{T}\right\vert < \left\vert{S}\right\vert$. Here, $\operatorname{im} \left({f}\right)$ denotes the image of $f$.

Thus $\operatorname{im} \left({f}\right) \ne S$, and so $f$ is not a bijection.

$\blacksquare$

Proof 2

Follows directly from Same Cardinality Bijective Injective Surjective.

Note

Some sources use this result as the property which defines a finite set.

Sources

• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 15 \theta$