Formulawise: RandomSet(S) =def (s)(S_i)(seS & S_i inc S --> p(seS_i)=p(S_i)).

The necessary and sufficient condition to make a set into a random set - for someone, in certain circumstances - is that one does not know anything in particular about any of the elements of the set.

One useful way to produce a random set R for a set S is to make names on slips of paper of the same size for all the elements in S; put these slips in a vase, bowl or urn; thoroughly shake it; and blindly select from it. This is how one samples from a population.