In
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
and its applications throughout
mathematics, a subclass is a
class contained in some other class in the same way that a
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
is a
set contained in some other set.
That is, given classes ''A'' and ''B'', ''A'' is a subclass of ''B''
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bi ...
every member of ''A'' is also a member of ''B''.
If ''A'' and ''B'' are sets, then of course ''A'' is also a subset of ''B''.
In fact, when using a definition of classes that requires them to be first-order definable, it is enough that ''B'' be a set; the
axiom of specification essentially says that ''A'' must then also be a set.
As with subsets, the
empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in oth ...
is a subclass of every class, and any class is a subclass of itself. But additionally, every class is a subclass of the class of all sets. Accordingly, the subclass relation makes the collection of all classes into a
Boolean lattice, which the subset relation does not do for the collection of all sets. Instead, the collection of all sets is an
ideal in the collection of all classes. (Of course, the collection of all classes is something larger than even a class!)
References
{{Reflist
Set theory