The MathResource
universal set
or universe, n. the domain with reference to which complementation in set theory is defined; the union of any set and its complement. Russell's paradox proves that this set cannot be all-inclusive, and in particular cannot include itself. Different formulations of set theory have adopted different devices to avoid this difficulty, but for practical purposes, such as Venn diagrams, it is sufficient to take the universal set to be some specific class large enough to include all the elements of any set relevant to the subject matter; for example, in a diagram of the relations between fish, mammals and aquatic animals, it may be sufficient to take animals as the universal set.