Cantor's paradox is that the set of all sets is its own one of these, and therefore must be bigger than itself.
A set S shatters A if this thing for A is equal to the set of intersections of elements of S with A. The existence of one of these for every set is one of the axioms of Zermelo-Fraenkel set theory. Cantor's paradox is that the set of all sets is its own one of these, and therefore must be bigger than itself. The singleton function, which takes a set X to this thing of X, maps every element of X to the singleton set containing it. If a set has cardinality n, then this thing for it has cardinality   2n. For ten points, identify this object associated with a set X which has as elements all of the subsets of X.
A set S shatters A if this thing for A is equal to the set of intersections of elements of S with A. The existence of one of these for every set is one of the axioms of Zermelo-Fraenkel set theory.
power set [prompt on descriptive answers like  set of subsets of a    certain set ]