Definition:Min Operation

Definition
Let $\left({S, \preceq}\right)$ be a totally ordered set.

The min operation is the binary operation on $\left({S, \preceq}\right)$ defined as:
 * $\forall x, y \in S: \min \left({x, y}\right) = \begin{cases}

x & : x \preceq y \\ y & : y \preceq x \end{cases}$

Notation
The notation $\min \left\{{x, y}\right\}$ is frequently seen for $\min \left({x, y}\right)$.

This emphasises that the operands of the min operation are undifferentiated as to order.