Cambridge Encyclopedia » Cambridge Encyclopedia Vol. 67

set - Describing sets, Set membership, Cardinality of a set, Subsets, Special sets, Unions, Intersections, Complements, Further reading

elements class universal written

In mathematics, a well-defined class of elements, ie a class where it is possible to tell exactly whether any one element does or does not belong to it. We can have the set of all even numbers, as every number is either even or not even, but we cannot have the set of all large numbers, as we do not know what is meant by ‘large’. The empty set ? is the set with no elements. The universal set ? or ? is the set of all elements, and the complement A? of a set A is the set of all elements in ? which are not in A. However, universal sets must also be defined carefully, else paradoxes result; one cannot speak of the set of all sets, for example. To do so would admit Russell's paradox (based on the set of all sets that are not members of themselves) which destroyed Frege's attempt to base all mathematics on logic. The intersection of two sets A and B (written A?B) is the set of all elements in both A and B. The union of two sets A and B (written A?B) is the set of all elements in either A or B or both.

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