### Basics Of Set Theory

##
**Set theory** is the branch of mathematical logic that studies sets,
which are collections of objects. Although any type of object can be
collected into a set, set theory is applied most often to objects that
are relevant to mathematics. The language of set theory can be used in
the definitions of nearly all mathematical objects.

## Basic concepts and notation

Set theory begins with a fundamental binary relation between an object*o*and a set

*A*. If

*o*is a

**member**(or

**element**) of

*A*, write

*o*∈

*A*. Since sets are objects, the membership relation can relate sets as well.

A derived binary relation between two sets is the subset relation, also called

**set inclusion**. If all the members of set

*A*are also members of set

*B*, then

*A*is a

**subset**of

*B*, denoted

*A*⊆

*B*. For example, {1,2} is a subset of {1,2,3} , but {1,4} is not. From this definition, it is clear that a set is a subset of itself; for cases where one wishes to rule out this, the term

**proper subset**is defined.

*A*is called a

**proper subset**of

*B*if and only if

*A*is a subset of

*B*, but

*B*is

**not**a subset of

*A*.

Just as arithmetic features binary operations on numbers, set theory features binary operations on sets. The:

**Union**of the sets*A*and*B*, denoted*A*∪*B*, is the set of all objects that are a member of*A*, or*B*, or both. The union of {1, 2, 3} and {2, 3, 4} is the set {1, 2, 3, 4} .**Intersection**of the sets*A*and*B*, denoted*A*∩*B*, is the set of all objects that are members of both*A*and*B*. The intersection of {1, 2, 3} and {2, 3, 4} is the set {2, 3} .**Set difference**of*U*and*A*, denoted*U*\*A*, is the set of all members of*U*that are not members of*A*. The set difference {1,2,3} \ {2,3,4} is {1} , while, conversely, the set difference {2,3,4} \ {1,2,3} is {4} . When*A*is a subset of*U*, the set difference*U*\*A*is also called the**complement**of*A*in*U*. In this case, if the choice of*U*is clear from the context, the notation*A*^{c}is sometimes used instead of*U*\*A*, particularly if*U*is a universal set as in the study of Venn diagrams.**Symmetric difference**of sets*A*and*B*, denoted*A*△*B*or*A*⊖*B*, is the set of all objects that are a member of exactly one of*A*and*B*(elements which are in one of the sets, but not in both). For instance, for the sets {1,2,3} and {2,3,4} , the symmetric difference set is {1,4} . It is the set difference of the union and the intersection, (*A*∪*B*) \ (*A*∩*B*) or (*A*\*B*) ∪ (*B*\*A*).**Cartesian product**of*A*and*B*, denoted*A*×*B*, is the set whose members are all possible ordered pairs (*a*,*b*) where*a*is a member of*A*and*b*is a member of*B*. The cartesian product of {1, 2} and {red, white} is {(1, red), (1, white), (2, red), (2, white)}.**Power set**of a set*A*is the set whose members are all possible subsets of*A*. For example, the power set of {1, 2} is { {}, {1}, {2}, {1,2} } .

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