Cardinality of Extensions of Function on Subset of Finite Set

Theorem

Let $m, n \in \Z_{>0}$ be (strictly) positive integers.

Let $S$ be a set with $m$ elements.

Let $T$ be a set with $n$ elements.

Let $A$ be a subset of $S$ with $r$ elements where $0 \le r < m$.

Let $f: A \to T$ be a mapping.

Then there are $n^{m - r}$ distinct extensions of $f$ to $S$.

Proof

Let $N$ denote the number of distinct extensions of $f$ to $S$.

The question is equivalent to asking the number of distinct mappings from $S \setminus A$ to $T$.

There are $m - r$ elements in $S \setminus A$.

Hence from Cardinality of Set of All Mappings:

$N = n^{m - r}$

$\blacksquare$

Sources

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 9$: Inverse Functions, Extensions, and Restrictions: Exercise $4$