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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}

2A

power set

all subsets of A

 

? (A)

power set

all subsets of A

 

A = B

equality

both sets have the same members

A={3,9,14}, B={3,9,14}, A=B

Ac

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

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 ? 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∈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

cardinality

the number of elements of set A

A={3,9,14}, #A=3

?

aleph

infinite cardinality

 

Ø

empty set

Ø = { }

C = {Ø}

U

universal set

set of all possible values

 

N0

natural numbers / whole numbers  set (with zero)

N0 = {0,1,2,3,4,...}

0 ∈ N0

N1

natural numbers / whole numbers  set (without zero)

N1 = {1,2,3,4,5,...}

6 ∈ N1

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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