complement is the empty or null set (); the union of a set and its complement is the
universal set.What are cardinaland ordinal numbersand finite setsin set theory?
Cardinal and ordinal numbers are used in reference to numbers: Ordinal numbers are
used to describe the position of objects or entities arranged in a certain sequence, such as
first, second, third, and so on; cardinal numbers are natural numbers, or 0, 1, 2, 3, and so
on. (For more information about cardinal and ordinal numbers, see “Math Basics.”)But cardinal numbers used in set theory describe the number of members in a set.
Both ordinal and cardinal numbers are further used to describe infinite sets and are
prefaced with “first ordinal” or “first cardinal” infinities. The first ordinal infinity
applies to the smallest number greater than any finite ordered set of natural numbers.
The first cardinal infinity applies to the number of all the natural numbers. (For more
about infinite sets, see below).A finite setis one that is not infinite. It can be numbered from 1 to n,for some
positive integer n. This number nis also called the set’s cardinal number; thus, for a
certain set A,the cardinality is denoted by card(A). There are a number of rules to car-
dinal numbers and finite sets. For example, if two sets bisect, then they are said to
have the same cardinality (or power). The empty set is considered to be a finite set,
with its set’s cardinal number being 0.What is a universal set?
A universal set actually applies to sets that are not universal but are chosen from a
specific type of entity, such as sets of numbers or letters. Thus, the set of all the ele-
ments in a set theory problem are collectively called the universal set. In reality, how-
ever, the “set of all things” does not exist because there is no largest or all-inclusive
set, and so the true universal set is not recognized in standard set theory.What is a subsetand proper subsetin set theory?
Simply put, a subsetis a portion of a set. If set Bis a subset of set A,then all elements
of set Bare also elements in set A. If Aand Bare equal, then both sets are subsets of
themselves; the empty set is also considered a subset of every other set. A proper sub-
setis a subset other than the set itself.When is a set a superset?
A superset is one that contains all the elements of a smaller set. For example, if Bis a
126 subset of A,then Ais a superset of B; in other words, Ais a superset of set Bif every