system of distinct representatives

Let $S=\{S_1,S_2,\mathrm{\dots }{S}_n\}$ be a finite collection of finite sets. A system of distinct representatives, or SDR, of $S$ is a set

$$x_1\in S_1,x_2\in S_2,\mathrm{\dots }{x}_n\in S_n$$

such that

$$x_i\ne x_j\text{ whenever }i\ne j$$

(i.e., each choice must be unique).