concepts in set theory

concepts in set theory

The aim of this entry is to present a list of the key objects and

concepts

used in

set theory

. Each entry in the list links
(or will link in the future) to the corresponding PlanetMath
entry where the object is presented in greater detail.
For convenience, this list also presents the encouraged
notation to use (at PlanetMath) for these objects.

•

set

•

set axioms

•

Venn diagrams

•

∅

, the

empty set

(also

{

}

or

∅

),

•

{

x

}

,

singleton

,

•

{

a

1

,

a

2

,

a

3

,

…

}

(

list form

), the set with

elements

a

1

,

a

2

,

a

3

,

…

,

•

x

∈

A

,

x

is an

element of

the set

A

,

•

A

∋

x

,

A

is a set containing

x

,

•

x

∉

A

,

x

is not an element of the set

A

,

•

A

∪

B

, union of sets

A

and

B

,

•

⋃

i

∈

I

A

i

, union of a family of sets

A

i

indexed by

elements of

I

,

•

A

⁢

∐

B

,

disjoint union

(or

A

⁢

∪

⋅

⁢

B

),

•

A

∩

B

,

intersection of sets

A

and

B

,

•

⋂

i

∈

I

A

i

,

intersection

of a family of sets

A

i

, indexed by elements in

I

,

•

A

∖

B

,

set difference

. An

alternative notation

for this is

A

-

B

, which should be avoided since in the context of

vector spaces

,

A

-

B

is
used for the set of all elements of the form

a

-

b

(see

Minkowski sum

(

http://planetmath.org/MinkowskiSum2

)),

•

A

/

∼

, set of

equivalence classes

in

A

determined by an

equivalence
relation

∼

in

A

,

•

[

a

]

, equivalence class in

A

/

∼

generated by

a

∈

A

,

•

A

∁

, set

complement

of

A

(where the ambient
set containing

A

is understood from context),

•

A

⁢

△

⁢

B

,

symmetric set difference

of

A

and

B

,

•

A

×

B

,

Cartesian product

of

A

and

B

,

•

∏

i

∈

I

A

i

, Cartesian product of the sets

A

i

(sometimes also

×

A

i

i

∈

I

),

•

id

X

,

identity mapping

X

→

X

,

•

𝒫

⁢

(

A

)

,

power set

of

A

(also

2

A

),

•

B

A

or

B

A

, the set of

functions

from

A

to

B

(rare outside of logic and set theory),

•

f

:

A

→

B

,

f

is a function having domain

A

and

codomain

B

,

•

card

⁡

(

A

)

,

cardinality

of

A

(also

♯

⁢

A

or

|

A

|

, which can be confused with the

absolute value

),

•

A

=

B

,

A

and

B

are equal (generally as sets; occasionally this notation is used to mean “

A

is canonically

isomorphic

to

B

”),

•

A

⊆

B

,

A

is a subset of

B

(or

A

⊂

B

, especially in set theory and logic),

•

A

⊊

B

,

A

is a

proper subset

(that is,

A

⊆

B

but

A

≠

B

; occasionally authors will use

A

⊂

B

to mean “proper subset”, conflicting with the above),

•

A

⊇

B

,

A

is a

superset

of

B

(with the same caveats as the previous entries).

•

discrete topology

Set builder notation

{

x

∈

A

⁢

such that

𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛

}

.

When

A

is

obvious

it may be omitted. Other symbols are also sometimes used in place of the words “such that”, for example

{

x

∈

A

∣

𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛

}

,

{

x

∈

A

:

𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛

}

,

{

x

∈

A

⁢

⋮

⁢

c

⁢

o

⁢

n

⁢

d

⁢

i

⁢

t

⁢

i

⁢

o

⁢

n

}

or

{

x

∈

A

⁢

s.t.

𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛

}

.

The reader should take care that if the objects under discussion are not just sets (say, groups or schemes) the

operations

may not be simple set operations, but rather their analogue in the relevant category. For example, the

product

of two groups is usually assigned a group law of a particular form, while the product of two schemes has “extra” points beyond those obtained from the Cartesian product of the schemes. Such conventions will normally be defined along with the category itself, although occasionally they will be an example of a general notion defined the same way in all categories (such as the categorical direct product).

Title

concepts in set theory

Canonical

name

ConceptsInSetTheory

Date of creation

2013-03-22 14:13:14

Last modified on

2013-03-22 14:13:14

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

54

Author

matte (1858)

Entry type

Topic

Classification

msc 03E99

Related topic

Set

Related topic

PolynomialFunction

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