Introduction to Set Theory
The Set theory is a field of mathematics that studies sets. Sets are the collections of objects. However any kind of object can be collected into a set, the set theory is applied most frequently to objects which are relevant to mathematics. The set theory language can be used in the definitions of almost all mathematical objects.
The study of set theory was started by Georg Cantor and Richard Dedekind in the year 1870. Subsequent to the discovery of paradoxes in naive set theory, the number of axiom systems was proposed in the early 20th century.
Set theory is generally employed as a foundational system for mathematics, mostly in the form of Zermelo-Fraenkel set theory with the axiom of choice. Beyond the introductory role the set theory is a field of mathematics in its own right with an active research community. Contemporary research into set theory which includes a various collection of topics, varying from the structure of the real number line to the study of the consistency of large cardinals.
Table of set theory symbols
Symbol
Symbol Name
Meaning / definition
Example
{ }
set
a collection of elements
A={3,7,9,14}, B={9,14,28}
A ∩ B
intersection
objects that belong to set A and set B
A ∩ B = {9,14}
A ∪ B
union
objects that belong to set A or set B
A ∪ B = {3,7,9,14,28}
A ⊆ B
subset
subset has fewer elements or equal to the set
{9,14,28} ⊆ {9,14,28}
A ⊂ B
proper subset / strict subset
subset has fewer elements than the set
{9,14} ⊂ {9,14,28}
A ⊄ B
not subset
left set not a subset of right set
{9,66} ⊄ {9,14,28}
A ⊇ B
superset
set A has more elements or equal to the set B
{9,14,28} ⊇ {9,14,28}
A ⊃ B
proper superset / strict superset
set A has more elements than set B
{9,14,28} ⊃ {9,14}
A ? B
not superset
set A is not a superset of set B
{9,14,28} ? {9,66}
2^{A}
power set
all subsets of A
? (A)
A = B
equality
both sets have the same members
A={3,9,14}, B={3,9,14}, A=B
A^{c}
complement
all the objects that do not belong to set A
A \ B
relative complement
objects that belong to A and not to B
A={3,9,14}, B={1,2,3}, A-B={9,14}
A - B
symmetric difference
objects that belong to A or B but not to their intersection
A={3,9,14}, B={1,2,3}, A ? B={1,2,9,14}
A={3,9,14}, B={1,2,3}, A ?B={1,2,9,14}
a∈A
element of
set membership
A={3,9,14}, 3 ∈ A
x∉A
not element of
no set membership
A={3,9,14}, 1 ∉ A
(a,b)
ordered pair
collection of 2 elements
A×B
Cartesian product
set of all ordered pairs from A and B
|A|
cardinality
the number of elements of set A
A={3,9,14}, |A|=3
#A
A={3,9,14}, #A=3
?
aleph
infinite cardinality
Ø
empty set
Ø = { }
C = {Ø}
U
universal set
set of all possible values
N_{0}
natural numbers / whole numbers set (with zero)
N_{0} = {0,1,2,3,4,...}
0 ∈ N_{0}
N_{1}
natural numbers / whole numbers set (without zero)
N_{1} = {1,2,3,4,5,...}
6 ∈ N_{1}
Z
integer numbers set
Z = {...-3,-2,-1,0,1,2,3,...}
-6 ∈ Z
Q
rational numbers set
Q = {x | x=a/b, a,b∈N}
2/6 ∈ Q
R
real numbers set
R = {x | -∞ < x <∞}
6.343434 ∈ R
C
complex numbers set
C = {z | z=a+bi, -∞<a<∞, -∞<b<∞}
6+2i ∈ C
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