# Cardinality of Set of All Mappings/Infinite Sets

## Theorem

Let $S$ and $T$ be sets such that either $S$ or $T$ is infinite.

The cardinality of the set of all mappings from $S$ to $T$ (that is, the total number of mappings from $S$ to $T$) is:

- $\left|{T^S}\right| = \left|{T}\right| ^ {\left|{S}\right|}$