Mathematical notation for choosing an element randomly from a set? I have a nonempty set $\mathcal{S}$ with finite number of elements. Is there any mathematical notation to randomly choose an element $x$ from the set $\mathcal{S}$ ? 
Each element in the set $\mathcal{S}$ has got equal probability (uniform probability) for getting chosen.
 A: I've seen a $S \xleftarrow{R} x$ used as a notation in protocol or crypto system descriptions, for drawing uniformly random elements $x$ from a set $S$.
A: $x \in_R S$ is a common way to denote that $x$ is chosen randomly from the set $S$. See, for example, this question.
A: In terms of logic, $S \neq \emptyset \implies \exists x[x \in S]$. The idea of picking an item from the set in a proof would involve assuming some particular instance of an item. An example of how that might work in a proof is as follows:
\begin{array}{l}
& S \neq \emptyset & \text{ Premise }\\
& \exists x[x \in S] & \text{ $S \neq \emptyset$ }\\
& a \in S & \text{ Assumption (some typical $a$) }\\
& \text{ ... Prove something, such as $P(a) = \frac{1}{|S|}$ ... }\\
& a \in S \land P(a) = \frac{1}{|S|} & \text{ $\land$ Introduction }\\
& \exists x[x \in S \land P(x) = \frac{1}{|S|}] & \text{ Existential Introduction }\\
\end{array}
The advantage of thinking of it this way is that the rule (Existential Elimination) for discharging such assumption is well defined. Basically, it involves ensuring that the conclusion drawn from the assumption is valid independently from the particular instance assumed.
