aarjaneiro/planetmath huggingface dataset:
product of non-empty set of non-empty sets is non-empty

product of non-empty set of non-empty sets is non-empty

In this entry, we show that the statement:

(*) the non-empty

generalized cartesian product

of non-empty sets is non-empty

is

equivalent

to the

axiom of choice

(AC).

Proposition 1

.

AC implies (*).

Proof.

Suppose

C

=

{

A

j

∣

j

∈

J

}

is a set of non-empty sets, with

J

≠

∅

. We want to show that

B

:=

∏

j

∈

J

A

j

is non-empty. Let

A

=

⋃

C

. Then, by AC, there is a

function

f

:

C

→

A

such that

f

⁢

(

X

)

∈

X

for every

X

∈

C

. Define

g

:

J

→

A

by

g

⁢

(

j

)

:=

f

⁢

(

A

j

)

. Then

g

∈

B

as a result,

B

is non-empty.
∎

Remark

. The statement that if

J

≠

∅

, then

B

≠

∅

implies

A

j

≠

∅

does not require AC: if

B

is non-empty, then there is a function

g

:

J

→

A

, and, as

J

≠

∅

,

g

≠

∅

, which means

A

≠

∅

, or that

A

j

≠

∅

for some

j

∈

J

.

Proposition 2

.

(*) implies AC.

Proof.

Suppose

C

is a set of non-empty sets. If

C

itself is empty, then the

choice function

is the

empty set

. So suppose that

C

is non-empty. We want to find a (choice) function

f

:

C

→

⋃

C

, such that

f

⁢

(

x

)

∈

x

for every

x

∈

C

. Index elements of

C

by

C

itself:

A

x

:=

x

for each

x

∈

C

. So

A

x

≠

∅

by

assumption

. Hence, by (*), the (non-empty)

cartesian product

B

of the

A

x

is non-empty. But an element of

B

is just a function

f

whose domain is

C

and whose codomain is the union of the

A

x

, or

⋃

C

, such that

f

⁢

(

A

x

)

∈

A

x

, which is precisely

f

⁢

(

x

)

∈

x

.
∎

Title

product of non-empty set of non-empty sets is non-empty

Canonical name

ProductOfNonemptySetOfNonemptySetsIsNonempty

Date of creation

2013-03-22 18:44:28

Last modified on

2013-03-22 18:44:28

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Derivation

Classification

msc 03E20

Generated on Thu Feb 8 20:14:13 2018 by

LaTeXML
proper subset

proper subset

Let

S

be a set and let

X

⊂

S

be a subset. We say

X

is a

proper subset

of

S

if

X

≠

S

.

Title

proper subset

Canonical name

ProperSubset

Date of creation

2013-03-22 11:53:28

Last modified on

2013-03-22 11:53:28

Owner

djao (24)

Last modified by

djao (24)

Numerical id

7

Author

djao (24)

Entry type

Definition

Classification

msc 03E20

Classification

msc 03B22

Classification

msc 03-00

Classification

msc 03-01

Generated on Thu Feb 8 20:14:19 2018 by

LaTeXML
properties of functions

properties of functions

Let

f

:

X

→

Y

be a

function

.
Let

(

A

i

)

i

∈

I

be a family of subsets of

X

,
and let

(

B

j

)

j

∈

J

be a family of subsets of

Y

,
where

I

and

J

are non-empty

index sets

.

Then, it is easy to prove, directly from

definitions

, that the following hold:

•

f

⁢

(

⋃

i

∈

I

A

i

)

=

⋃

i

∈

I

f

⁢

(

A

i

)

(i.e., the image of a union is the union of the images)

•

f

⁢

(

⋂

i

∈

I

A

i

)

⊆

⋂

i

∈

I

f

⁢

(

A

i

)

(i.e., the image of an

intersection

is

contained

in the intersection of the images)

•

A

⊆

f

-

1

⁢

(

f

⁢

(

A

)

)

for any

A

⊆

X

(where

f

-

1

⁢

(

f

⁢

(

A

)

)

is the

inverse image

of

f

⁢

(

A

)

)

•

f

⁢

(

f

-

1

⁢

(

B

)

)

⊆

B

for any

B

⊆

Y

•

f

-

1

⁢

(

Y

∖

B

)

=

X

∖

f

-

1

⁢

(

B

)

for any

B

⊆

Y

•

f

-

1

⁢

(

⋃

j

∈

J

B

j

)

=

⋃

j

∈

J

f

-

1

⁢

(

B

j

)

(the inverse image of a union is the union of the inverse images)

•

f

-

1

⁢

(

⋂

j

∈

J

B

j

)

=

⋂

j

∈

J

f

-

1

⁢

(

B

j

)

(the inverse image of an intersection is the intersection of the inverse images)

•

f

⁢

(

f

-

1

⁢

(

B

)

)

=

B

for every

B

⊆

Y

if and only if

f

is

surjective

.

For more

properties

related specifically to inverse images, see the inverse image (

http://planetmath.org/InverseImage

) entry.

Further, the following conditions are

equivalent

(for more, see the entry on

injective functions

):

•

f

is injective

•

f

⁢

(

S

∩

T

)

=

f

⁢

(

S

)

∩

f

⁢

(

T

)

for all

S

,

T

⊆

X

•

f

-

1

⁢

(

f

⁢

(

S

)

)

=

S

for all

S

⊆

X

•

f

⁢

(

S

)

∩

f

⁢

(

T

)

=

∅

for all

S

,

T

⊆

X

such that

S

∩

T

=

∅

•

f

⁢

(

S

∖

T

)

=

f

⁢

(

S

)

∖

f

⁢

(

T

)

for all

S

,

T

⊆

X

Title

properties of functions

Canonical

name

PropertiesOfFunctions

Date of creation

2013-03-22 14:59:54

Last modified on

2013-03-22 14:59:54

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

20

Author

yark (2760)

Entry type

Result

Classification

msc 03E20

Related topic

PropertiesOfAFunction

Generated on Thu Feb 8 20:14:34 2018 by

LaTeXML
range

range

Let

R

be a

binary relation

. Then the set of all

y

such that

x

⁢

R

⁢

y

for some

x

is called the

range

of

R

. That is, the range of

R

is the set of all second coordinates in the

ordered pairs

of

R

.

In of

functions

, this means that the range of a function is the full set of values it can take on (the outputs), given the full set of

parameters

(the inputs). Note that the range is a subset of the codomain.

Title

range

Canonical

name

Range

Date of creation

2013-03-22 12:00:51

Last modified on

2013-03-22 12:00:51

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

6

Author

akrowne (2)

Entry type

Definition

Classification

msc 03E20

Generated on Thu Feb 8 20:15:07 2018 by

LaTeXML
reflexive relation

reflexive relation

A

relation

ℛ

on a set

A

is

reflexive

if and only if

a

⁢

ℛ

⁢

a

for all

a

∈

A

.

For example, let

A

=

{

1

,

2

,

3

}

.
Then

{

(

1

,

1

)

,

(

2

,

2

)

,

(

3

,

3

)

,

(

1

,

3

)

,

(

3

,

2

)

}

is a reflexive relation on

A

,
because it contains

(

a

,

a

)

for all

a

∈

A

.
However,

{

(

1

,

1

)

,

(

2

,

2

)

,

(

2

,

3

)

,

(

3

,

1

)

}

is not reflexive
because it does not contain

(

3

,

3

)

.

On a

finite set

with

n

elements

there are

2

n

2

relations,
of which

2

n

2

-

n

are reflexive.

Title

reflexive relation

Canonical name

ReflexiveRelation

Date of creation

2013-03-22 12:15:36

Last modified on

2013-03-22 12:15:36

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

17

Author

yark (2760)

Entry type

Definition

Classification

msc 03E20

Related topic

Symmetric

Related topic

Transitive3

Related topic

Antisymmetric

Related topic

Irreflexive

Defines

reflexivity

Defines

reflexive

Generated on Thu Feb 8 20:15:09 2018 by

LaTeXML
restriction of a function

restriction of a function

Definition

Let

f

:

X

→

Y

be a

function

from a set

X

to a set

Y

.
If

A

is a subset of

X

,
then the

restriction

of

f

to

A

is the function

f

|

A

:

A

→

Y

x

↦

f

⁢

(

x

)

.

Some authors write

f

↾

A

instead of

f

|

A

.

Properties

If

A

⊆

X

, and

B

⊆

Y

, then

(

f

|

A

)

-

1

⁢

(

B

)

=

A

∩

f

-

1

⁢

(

B

)

.

Title

restriction of a function

Canonical name

RestrictionOfAFunction

Date of creation

2013-03-22 13:43:05

Last modified on

2013-03-22 13:43:05

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

14

Author

yark (2760)

Entry type

Definition

Classification

msc 03E20

Synonym

restriction

Related topic

Pullback2

Related topic

ExtensionOfAFunction

Generated on Thu Feb 8 20:15:11 2018 by

LaTeXML
right function notation

right function notation

We are said to be using
if we write

functions

to the right of their

arguments

.
That is, if

α

:

X

→

Y

is a function and

x

∈

X

,
then

x

⁢

α

is the image of

x

under

α

.

Furthermore, if we have a function

β

:

Y

→

Z

,
then we write the

composition

of the two functions
as

α

⁢

β

:

X

→

Z

,
and the image of

x

under the composition
as

x

⁢

α

⁢

β

=

x

⁢

(

α

⁢

β

)

=

(

x

⁢

α

)

⁢

β

.

Compare this to

left function notation

.

Title

right function notation

Canonical name

RightFunctionNotation

Date of creation

2013-03-22 12:09:22

Last modified on

2013-03-22 12:09:22

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

6

Author

antizeus (11)

Entry type

Definition

Classification

msc 03E20

Synonym

right notation

Generated on Thu Feb 8 20:15:13 2018 by

LaTeXML
Schröder Bernstein Theorem: Proof

Schröder Bernstein Theorem: Proof

Let

A

and

B

be two

nonempty sets

; and let there be, in

addition

, two one-one

functions

f

:

A

↣

B

and

g

:

B

↣

A

. We propose to show that

A

and

B

are equinumerous i.e., they are in one to one correspondence.

Consider the notation:

g

-

1

⁢

(

x

)

,

if

x

∈

g

⁢

(

B

)

f

-

1

⁢

(

g

-

1

⁢

(

x

)

)

,

if

g

-

1

⁢

(

x

)

∈

f

⁢

(

A

)

g

-

1

⁢

(

f

-

1

⁢

(

g

-

1

⁢

(

x

)

)

)

,

if

f

-

1

⁢

(

g

-

1

⁢

(

x

)

)

∈

g

⁢

(

B

)

…

.

.

Define, for each

x

∈

A

, the order of it, denoted by

∘

(

x

)

, to be the number of such

preimage

(s) which exist. In a similer way, we’d be able to define the order of an

element

y

∈

B

, i.e., by considering the

sequence

f

-

1

⁢

(

y

)

,

g

-

1

⁢

(

f

-

1

⁢

(

y

)

)

,

…

.

Now define, for each

x

∈

A

,

ϕ

⁢

(

x

)

:

=

f

(

x

)

,

∘

(

x

)

=

∞

=

f

(

x

)

,

∘

(

x

)

=

2

n

,

for some

n

∈

ω

=

b

,

∘

(

x

)

=

2

n

+

1

,

i

.

e

.

,

∃

b

∈

B

:

g

(

b

)

=

x

Notice that if the order is

infinite

,

ϕ

⁢

(

x

)

=

f

⁢

(

x

)

is also infinite. Because, otherwise

x

would have to have a finite order. On the other hand, if

y

∈

B

and

∘

(

y

)

is infinite, then

f

-

1

⁢

(

y

)

exists and has an

infinite order

; call the latter one

x

. This means,

ϕ

maps the infinite order elements of

A

bijectively onto the infinite order elements of

B

.

Next, if

∘

(

x

)

=

2

n

, then the order of

f

⁢

(

x

)

is sheer

2

⁢

n

+

1

. Similer to the above para, if for

y

∈

B

, the order is

2

⁢

n

+

1

, as the order is non-zero,

f

-

1

⁢

(

y

)

exists and it must have order

2

⁢

n

. To formally show it you need a tedious inductive reasoning!

Last, if

∘

(

x

)

=

2

n

+

1

, then the order is

≥

1

, and so,

g

-

1

⁢

(

x

)

exists and the order of

g

-

1

⁢

(

x

)

is sheer

2

⁢

n

(looking upon

g

-

1

⁢

(

x

)

as an element of

B

).

Conversely

, similer to above, if there is an element

y

of order

2

⁢

n

in

B

, take

x

=

g

⁢

(

y

)

and the order of

x

is indeed

2

⁢

n

+

1

. All that you need to convince a sceptic is a long, tedious, involved

induction

!!

What we learn from what precedes is that the one-one function

ϕ

maps the infinite order elements onto infinite order elements, odd order onto odd order, and even order onto even order; a simple set theory reveals that

ϕ

is a one-one map from

A

onto

B

. This

completes

the proof.

Title

Schröder Bernstein Theorem: Proof

Canonical name

SchroderBernsteinTheoremProof

Date of creation

2013-03-22 16:40:36

Last modified on

2013-03-22 16:40:36

Owner

sauravbhaumik (15615)

Last modified by

sauravbhaumik (15615)

Numerical id

10

Author

sauravbhaumik (15615)

Entry type

Proof

Classification

msc 03E20

Generated on Thu Feb 8 20:15:18 2018 by

LaTeXML
set difference

set difference

Definition

Let

A

and

B

be sets.
The

set difference

(or simply

difference

)
between

A

and

B

(in that order)
is the set of all elements of

A

that are not in

B

.
This set is denoted by

A

∖

B

or

A

-

B

(or occasionally

A

∼

B

). So we have

A

∖

B

=

{

x

∈

A

∣

x

∉

B

}

.

Venn diagram

showing

A

∖

B

in blue

Properties

Here are some

properties

of the set difference

operation

:

1.

If

A

is a set, then

A

∖

∅

=

A

and

A

∖

A

=

∅

=

∅

∖

A

.

2.

If

A

and

B

are sets, then

B

∖

(

A

∩

B

)

=

B

∖

A

.

3.

If

A

and

B

are subsets of a set

X

, then

A

∖

B

=

A

∩

B

∁

and

(

A

∖

B

)

∁

=

A

∁

∪

B

,

where

∁

denotes

complement

in

X

.

4.

If

A

,

B

,

C

and

D

are sets, then

(

A

∖

B

)

∩

(

C

∖

D

)

=

(

A

∩

C

)

∖

(

B

∪

D

)

.

Remark

As noted above, the set difference is sometimes written as

A

-

B

.
However, if

A

and

B

are sets in a

vector space

(or, more generally, a module (

http://planetmath.org/Module

)),
then

A

-

B

is commonly used to denote the set

A

-

B

=

{

a

-

b

∣

a

∈

A

,

b

∈

B

}

rather than the set difference.

Title

set difference

Canonical

name

SetDifference

Date of creation

2013-03-22 11:59:38

Last modified on

2013-03-22 11:59:38

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

33

Author

yark (2760)

Entry type

Definition

Classification

msc 03E20

Synonym

difference between sets

Synonym

difference

Related topic

SymmetricDifference

Related topic

InverseImageCommutesWithSetOperations

Related topic

Difference2

Generated on Thu Feb 8 20:15:20 2018 by

LaTeXML
sum and product and quotient of functions

sum and product and quotient of functions

Let

A

be a set and

M

a left

R

-module.  If

f

:

A

→

M

and

g

:

A

→

M

,  then one may define the

sum of functions

f

and

g

as the following

function

f

+

g

:

A

→

M

:

(

f

+

g

)

⁢

(

x

)

:=

f

⁢

(

x

)

+

g

⁢

(

x

)

∀

x

∈

A

If

r

is any

element of

the ring

R

, then the

scalar-multiplied function

r

⁢

f

:

A

→

M

is defined as

(

r

⁢

f

)

⁢

(

x

)

:=

r

⋅

f

⁢

(

x

)

∀

x

∈

A

.

Let

A

again be a set and

K

a field or a skew field.  If

f

:

A

→

K

and

g

:

A

→

K

,  then one can define the

product of functions

f

and

g

as the function

f

⁢

g

:

A

→

K

as follows:

(

f

⁢

g

)

⁢

(

x

)

:=

f

⁢

(

x

)

⋅

g

⁢

(

x

)

∀

x

∈

A

The

quotient of functions

f

and

g

is the function

f

g

:

{

a

∈

A

⋮

g

(

a

)

≠

0

}

→

K

defined as

f

g

(

x

)

:=

f

⁢

(

x

)

g

⁢

(

x

)

∀

x

∈

A

∖

{

a

∈

A

⋮

g

(

a

)

=

0

}

.

In particular, the incremental quotient of functions

f

⁢

(

y

)

-

f

⁢

(

x

)

y

-

x

, as

y

tends to

x

, gave rise to the important

concept

of

derivative

. As another example, we can with a conscience say that the

tangent

(

http://planetmath.org/TrigonometricFunction

) function is the

quotient

of the sine (

http://planetmath.org/TrigonometricFunction

) and the

cosine functions

.

Title

sum and product and quotient of functions

Canonical

name

SumAndProductAndQuotientOfFunctions

Date of creation

2013-03-22 17:44:24

Last modified on

2013-03-22 17:44:24

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

12

Author

pahio (2872)

Entry type

Definition

Classification

msc 03E20

Related topic

DirectSumOfEvenoddFunctionsExample

Related topic

LimitRulesOfFunctions

Related topic

PolynomialFunction

Related topic

ProofOfLimitRuleOfProduct

Related topic

ContinuousDerivativeImpliesBoundedVariation

Related topic

PropertiesOfRiemannStieltjesIntegral

Related topic

InfimumAndSupremumOfSumAndProduct

Related topic

PropertiesOfVectorValuedFunctio

Defines

sum of functions

Defines

product of functions

Defines

quotient of functions

Defines

scalar-multiplied function

Generated on Thu Feb 8 20:15:23 2018 by

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symmetric difference

symmetric difference

The

symmetric difference

between two sets

A

and

B

, written

A

⁢

△

⁢

B

, is the set of all

x

such that either

x

∈

A

or

x

∈

B

but not both. In other words,

A

⁢

△

⁢

B

:=

(

A

∪

B

)

∖

(

A

∩

B

)

.

The

Venn diagram

for the symmetric difference of two sets

A

,

B

, represented by the two discs, is illustrated below, in light red:

Properties

Suppose that

A

,

B

, and

C

are sets.

•

A

⁢

△

⁢

B

=

(

A

∖

B

)

∪

(

B

∖

A

)

.

•

A

⁢

△

⁢

B

=

A

c

⁢

△

⁢

B

c

, where the

superscript

c

denotes taking

complements

.

•

Note that for any set

A

, the symmetric difference

satisfies

A

⁢

△

⁢

A

=

∅

and

A

⁢

△

⁢

∅

=

A

.

•

The symmetric difference operator is

commutative

since

A

⁢

△

⁢

B

=

(

A

∖

B

)

∪

(

B

∖

A

)

=

(

B

∖

A

)

∪

(

A

∖

B

)

=

B

⁢

△

⁢

A

.

•

The symmetric difference

operation

is

associative

:

(

A

⁢

△

⁢

B

)

⁢

△

⁢

C

=

A

⁢

△

⁢

(

B

⁢

△

⁢

C

)

. This means that we may drop the parentheses without any ambiguity, and we can talk about the symmetric difference of

multiple

sets.

•

Let

A

1

,

…

,

A

n

be sets. The symmetric difference of these sets is written

n

△

i

=

1

⁢

A

i

.

In general, an

element

will be in the symmetric difference of several sets iff it is in an

odd number

of the sets.

It is worth noting that these

properties

show that the symmetric difference operation can be used as a group law to define an

abelian group

on the

power set

of some set.

Finally, we note that

intersection

distributes over

the symmetric difference operator:

A

∩

(

B

⁢

△

⁢

C

)

=

(

A

∩

B

)

⁢

△

⁢

(

A

∩

C

)

,

giving us that the power set of a given

fixed set

can be made into a

Boolean ring

using symmetric difference as

addition

, and intersection as

multiplication

.

Title

symmetric difference

Canonical

name

SymmetricDifference

Date of creation

2013-03-22 11:59:41

Last modified on

2013-03-22 11:59:41

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

20

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03E20

Synonym

set symmetric difference

Synonym

symmetric set difference

Synonym

symmetric difference between sets

Related topic

SetDifference

Related topic

ProofOfTheAssociativityOfTheSymmetricDifferenceOperator

Defines

symmetric difference operator

Generated on Thu Feb 8 20:15:25 2018 by

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symmetric relation

symmetric relation

A

relation

ℛ

on a set

A

is

symmetric

if and only if
whenever

x

⁢

ℛ

⁢

y

for some

x

,

y

∈

A

then also

y

⁢

ℛ

⁢

x

.

An example of a

symmetric relation

on

{

a

,

b

,

c

}

is

{

(

a

,

a

)

,

(

c

,

b

)

,

(

b

,

c

)

,

(

a

,

c

)

,

(

c

,

a

)

}

.
One relation that is not symmetric is

ℛ

=

{

(

b

,

b

)

,

(

a

,

b

)

,

(

b

,

a

)

,

(

c

,

b

)

}

,
because

(

c

,

b

)

∈

ℛ

but

(

b

,

c

)

∉

ℛ

.

On a

finite set

with

n

elements

there are

2

n

2

relations,
of which

2

n

2

+

n

2

are symmetric.

A relation

ℛ

that is both symmetric and

antisymmetric

has the property that

x

⁢

ℛ

⁢

y

implies

x

=

y

.
On a finite set with

n

elements there are only

2

n

such relations.

Title

symmetric relation

Canonical name

SymmetricRelation

Date of creation

2013-03-22 12:15:39

Last modified on

2013-03-22 12:15:39

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

21

Author

yark (2760)

Entry type

Definition

Classification

msc 03E20

Related topic

Reflexive

Related topic

Transitive3

Related topic

Antisymmetric

Defines

symmetry

Defines

symmetric

Generated on Thu Feb 8 20:15:40 2018 by

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the inverse image commutes with set operations

the inverse image commutes with set operations

Theorem

.

Let

f

be a

mapping

from

X

to

Y

. If

{

B

i

}

i

∈

I

is
a (possibly

uncountable

)

collection

of subsets in

Y

, then
the following

relations

hold for the

inverse image

:

(1)

f

-

1

⁢

(

⋃

i

∈

I

B

i

)

=

⋃

i

∈

I

f

-

1

⁢

(

B

i

)

(2)

f

-

1

⁢

(

⋂

i

∈

I

B

i

)

=

⋂

i

∈

I

f

-

1

⁢

(

B

i

)

If

A

and

B

are subsets in

Y

, then we also have:

(3)

For the set

complement

,

(

f

-

1

⁢

(

A

)

)

∁

=

f

-

1

⁢

(

A

∁

)

.

(4)

For the

set difference

,

f

-

1

⁢

(

A

∖

B

)

=

f

-

1

⁢

(

A

)

∖

f

-

1

⁢

(

B

)

.

(5)

For the

symmetric difference

,

f

-

1

⁢

(

A

△

B

)

=

f

-

1

⁢

(

A

)

△

f

-

1

⁢

(

B

)

.

Proof.

For part (1), we have

f

-

1

⁢

(

⋃

i

∈

I

B

i

)

=

{

x

∈

X

∣

f

⁢

(

x

)

∈

⋃

i

∈

I

B

i

}

=

{

x

∈

X

∣

f

⁢

(

x

)

∈

B

i

⁢

for some

⁢

i

∈

I

}

=

⋃

i

∈

I

{

x

∈

X

∣

f

⁢

(

x

)

∈

B

i

}

=

⋃

i

∈

I

f

-

1

⁢

(

B

i

)

.

Similarly, for part (2), we have

f

-

1

⁢

(

⋂

i

∈

I

B

i

)

=

{

x

∈

X

∣

f

⁢

(

x

)

∈

⋂

i

∈

I

B

i

}

=

{

x

∈

X

∣

f

⁢

(

x

)

∈

B

i

⁢

for all

⁢

i

∈

I

}

=

⋂

i

∈

I

{

x

∈

X

∣

f

⁢

(

x

)

∈

B

i

}

=

⋂

i

∈

I

f

-

1

⁢

(

B

i

)

.

For the set complement, suppose

x

∉

f

-

1

⁢

(

A

)

. This is

equivalent

to

f

⁢

(

x

)

∉

A

, or

f

⁢

(

x

)

∈

A

∁

, which is equivalent to

x

∈

f

-

1

⁢

(

A

∁

)

. Since the set difference

A

∖

B

can be
written as

A

∩

B

c

, part (4) follows from parts (2) and
(3). Similarly, since

A

△

B

=

(

A

∖

B

)

∪

(

B

∖

A

)

,
part (5) follows from parts (1) and
(4).

□

Title

the

inverse image commutes with set operations

Canonical name

TheInverseImageCommutesWithSetOperations

Date of creation

2013-03-22 13:35:24

Last modified on

2013-03-22 13:35:24

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

11

Author

matte (1858)

Entry type

Proof

Classification

msc 03E20

Related topic

SetDifference

Generated on Thu Feb 8 20:15:44 2018 by

LaTeXML
transformation

transformation

Synonym of

mapping

and

function

. Often used to refer to mappings where the domain and
codomain are the same set, i.e. one can compose a

transformation

with itself. For example, when one
speaks of

transformation of a space

, one refers to some deformation/rearrangement of that space.

Title

transformation

Canonical name

Transformation

Date of creation

2013-03-22 12:20:19

Last modified on

2013-03-22 12:20:19

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

6

Author

rmilson (146)

Entry type

Definition

Classification

msc 03E20

Synonym

mapping function

Related topic

Invariant

Related topic

Mapping

Related topic

Fixed

Generated on Thu Feb 8 20:15:46 2018 by

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transitive

transitive

Let

A

be a set.

A

is said to be

transitive

if whenever

x

∈

A

then

x

⊆

A

.

Equivalently,

A

is transitive if whenever

x

∈

A

and

y

∈

x

then

y

∈

A

.

Title

transitive

Canonical

name

Transitive

Date of creation

2013-03-22 12:13:00

Last modified on

2013-03-22 12:13:00

Owner

Evandar (27)

Last modified by

Evandar (27)

Numerical id

6

Author

Evandar (27)

Entry type

Definition

Classification

msc 03E20

Synonym

transitive set

Related topic

TransitiveClosure

Generated on Thu Feb 8 20:15:47 2018 by

LaTeXML
transitive closure

transitive closure

The

transitive closure

of a set

X

is the smallest

transitive set

tc

⁡

(

X

)

such that

X

⊆

tc

⁡

(

X

)

.

The transitive closure of a set can be constructed as follows:

Define a

function

f

on

ω

by

f

⁢

(

0

)

=

X

and

f

⁢

(

n

+

1

)

=

⋃

f

⁢

(

n

)

tc

⁡

(

X

)

=

⋃

n

<

ω

f

⁢

(

n

)

Title

transitive closure

Canonical name

TransitiveClosure

Date of creation

2013-03-22 13:04:33

Last modified on

2013-03-22 13:04:33

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

4

Author

Henry (455)

Entry type

Definition

Classification

msc 03E20

Related topic

Transitive

Generated on Thu Feb 8 20:15:49 2018 by

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