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 \in 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 \subseteq 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 $A \cup B$ , 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 $A \cap B$ , 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 $A \times B$ , 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 ${ {}, {1}, {2}, {1,2} }$ .

Some basic sets of central importance are the empty set (the unique set containing no elements), the set of natural numbers, and the set of real numbers.

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Basics Of Set Theory

Basic concepts and notation

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