aarjaneiro/planetmath huggingface dataset:
properties of cardinal numbers

properties of cardinal numbers

Theorem

.

Let

(

c

γ

)

γ

∈

Γ

be an indexed family of

cardinal numbers

indexed by

a nonempty

index set

Γ

. Also, let

(

Γ

δ

)

δ

∈

Δ

be an arbitrary indexed

partition

of the index set. Then we have the following properties:

1.

Associative

Laws.

∑

γ

∈

Γ

c

γ

=

∑

δ

∈

Δ

∑

γ

∈

Γ

δ

c

γ

and

∏

γ

∈

Γ

c

γ

=

∏

δ

∈

Δ

∏

γ

∈

Γ

δ

c

γ

.

2.

Commutative Laws

.

Let

π

:

Γ

→

Γ

be a partition. Then

∑

γ

∈

Γ

c

γ

=

∑

γ

∈

Γ

c

π

⁢

(

γ

)

and

∏

γ

∈

Γ

c

γ

=

∏

γ

∈

Γ

c

π

⁢

(

γ

)

.

3.

Distributive Laws

.

Let

a

be any arbitrary

infinite

cardinal number. Then

a

⁢

(

∑

γ

∈

Γ

c

γ

)

=

∑

γ

∈

Γ

a

⁢

c

γ

Title

properties of cardinal numbers

Canonical name

PropertiesOfCardinalNumbers

Date of creation

2013-03-22 16:08:29

Last modified on

2013-03-22 16:08:29

Owner

gilbert_51126 (14238)

Last modified by

gilbert_51126 (14238)

Numerical id

7

Author

gilbert_51126 (14238)

Entry type

Theorem

Classification

msc 03-00

Generated on Thu Feb 8 19:20:00 2018 by

LaTeXML
Ψ is surjective if and only if Ψ^∗ is injective

Ψ

is surjective if and only if

Ψ

∗

is injective

Suppose

X

is a set and

V

is a

vector space

over a field

F

.
Let us denote by

M

⁢

(

X

,

V

)

the set of

mappings

from

X

to

V

.
Now

M

⁢

(

X

,

V

)

is again a vector space if we equip it with

pointwise

multiplication

and

addition

. In detail,
if

f

,

g

∈

M

⁢

(

X

,

V

)

and

μ

,

λ

∈

F

, we set

μ

⁢

f

+

λ

⁢

g

:

x

↦

μ

⁢

f

⁢

(

x

)

+

λ

⁢

g

⁢

(

x

)

.

Next, let

Y

be another set, let

Ψ

:

X

→

Y

is a mapping,
and let

Ψ

∗

:

M

⁢

(

Y

,

V

)

→

M

⁢

(

X

,

V

)

be the

pullback

of

Ψ

as defined in this (

http://planetmath.org/Pullback2

) entry.

Proposition 1

.

1.

Ψ

∗

is linear.

2.

If

V

is not the

zero vector space

, then

Ψ

is

surjective

if and only if

Ψ

∗

is

injective

.

Proof.

First, suppose

f

,

g

∈

M

⁢

(

Y

,

V

)

,

μ

,

λ

∈

F

, and

x

∈

X

. Then

Ψ

∗

⁢

(

μ

⁢

f

+

λ

⁢

g

)

⁢

(

x

)

=

(

μ

⁢

f

+

λ

⁢

g

)

⁢

(

Ψ

⁢

(

x

)

)

=

μ

⁢

f

∘

Ψ

⁢

(

x

)

+

λ

⁢

g

∘

Ψ

⁢

(

x

)

=

(

μ

⁢

Ψ

∗

⁢

(

f

)

+

λ

⁢

Ψ

∗

⁢

(

g

)

)

⁢

(

x

)

,

so

Ψ

∗

⁢

(

μ

⁢

f

+

λ

⁢

g

)

=

μ

⁢

Ψ

∗

⁢

(

f

)

+

λ

⁢

Ψ

∗

⁢

(

g

)

,
and

Ψ

∗

is linear.
For the second claim, suppose

Ψ

is surjective,

f

∈

M

⁢

(

Y

,

V

)

,
and

Ψ

∗

⁢

(

f

)

=

0

. If

y

∈

Y

, then for some

x

∈

X

, we have

Ψ

⁢

(

x

)

=

y

, and

f

⁢

(

y

)

=

f

∘

Ψ

⁢

(

x

)

=

Ψ

∗

⁢

(

f

)

⁢

(

x

)

=

0

, so

f

=

0

.
Hence, the kernel of

Ψ

∗

is zero, and

Ψ

∗

is an
injection.
On the other hand, suppose

Ψ

∗

is a injection, and

Ψ

is not a surjection. Then for some

y

′

∈

Y

, we have

y

′

∉

Ψ

⁢

(

X

)

. Also, as

V

is not the zero vector space, we can
find a non-zero vector

v

∈

V

, and define

f

∈

M

⁢

(

Y

,

V

)

as

f

⁢

(

y

)

=

{

v

,

if

⁢

y

=

y

′

,

0

,

if

⁢

y

≠

y

′

,

y

∈

Y

.

Now

f

∘

Ψ

⁢

(

x

)

=

0

for all

x

∈

X

, so

Ψ

∗

⁢

f

=

0

,
but

f

≠

0

. ∎

Title

Ψ

is surjective if and only if

Ψ

∗

is injective

Canonical name

PsiIsSurjectiveIfAndOnlyIfPsiastIsInjective

Date of creation

2013-03-22 14:36:03

Last modified on

2013-03-22 14:36:03

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

6

Author

matte (1858)

Entry type

Theorem

Classification

msc 03-00

Generated on Thu Feb 8 19:20:02 2018 by

LaTeXML
pullback

pullback

Definition

Suppose

X

,

Y

,

Z

are sets, and we have maps

f

:

Y

→

Z

,

Φ

:

X

→

Y

.

Then the

pullback

of

f

under

Φ

is the

mapping

Φ

∗

⁢

f

:

X

→

Z

,

x

↦

(

f

∘

Φ

)

⁢

(

x

)

.

Let us denote by

M

⁢

(

X

,

Y

)

the set of all mappings

f

:

X

→

Y

.
We then see that

Φ

∗

is a mapping

M

⁢

(

Y

,

Z

)

→

M

⁢

(

X

,

Z

)

.
In other words,

Φ

∗

pulls back the set where

f

is
defined on from

Y

to

X

. This is illustrated in the below diagram.

\xymatrix

⁢

X

⁢

\ar

⁢

[

r

]

Φ

⁢

\ar

⁢

[

d

⁢

r

]

Φ

∗

⁢

f

⁢

&

⁢

Y

⁢

\ar

⁢

[

d

]

f

⁢

&

⁢

Z

0.0.1

Properties

1.

For any set

X

,

(

id

X

)

∗

=

id

M

⁢

(

X

,

X

)

.

2.

Suppose we have maps

Φ

:

X

→

Y

,

Ψ

:

Y

→

Z

between sets

X

,

Y

,

Z

. Then

(

Ψ

∘

Φ

)

∗

=

Φ

∗

∘

Ψ

∗

.

3.

If

Φ

:

X

→

Y

is a

bijection

, then

Φ

∗

is a bijection and

(

Φ

∗

)

-

1

=

(

Φ

-

1

)

∗

.

4.

Suppose

X

,

Y

are sets with

X

⊂

Y

.
Then we have the

inclusion map

ι

:

X

↪

Y

, and
for any

f

:

Y

→

Z

, we have

ι

∗

⁢

f

=

f

|

X

,

where

f

|

X

is the

restriction

(

http://planetmath.org/RestrictionOfAFunction

) of

f

to

X

.

Title

pullback

Canonical

name

Pullback

Date of creation

2013-03-22 13:50:04

Last modified on

2013-03-22 13:50:04

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

14

Author

matte (1858)

Entry type

Definition

Classification

msc 03-00

Related topic

InclusionMapping

Related topic

RestrictionOfAFunction

Related topic

PullbackOfAKForm

Generated on Thu Feb 8 19:20:05 2018 by

LaTeXML
Russell’s paradox

Russell’s paradox

Suppose that for any coherent

proposition

P

⁢

(

x

)

, we can construct a set

{

x

:

P

⁢

(

x

)

}

.
Let

S

=

{

x

:

x

∉

x

}

. Suppose

S

∈

S

; then, by

definition

,

S

∉

S

. Likewise, if

S

∉

S

, then by definition

S

∈

S

. Therefore, we have a

contradiction

.
Bertrand Russell gave this

paradox

as an example of how a purely intuitive

set theory

can be

inconsistent

. The regularity axiom, one of the

Zermelo-Fraenkel axioms

, was devised to avoid this paradox by prohibiting self-swallowing sets.

An

interpretation

of

Russell paradox

without any

formal language

of set theory could be stated like “If the barber shaves all those who do not themselves shave, does he shave himself?”. If you answer himself that is false since he only shaves all those who do not themselves shave. If you answer someone else that is also false because he shaves all those who do not themselves shave and in this case he is part of that set since he does not shave himself. Therefore we have a contradiction.

Remark

. Russell’s paradox is the result of an axiom (due to Frege) in set theory, now obsolete, known as the

axiom of (unrestricted)

comprehension

, which states: if

ϕ

is a

predicate

in the

language

of set theory, then there is a set that contains exactly those

elements

x

such that

ϕ

⁢

(

x

)

. In other words,

{

x

∣

ϕ

⁢

(

x

)

}

is a set. So if we take

ϕ

⁢

(

x

)

to be

x

∉

x

, we arrive at Russell’s paradox.

Title

Russell’s paradox

Canonical

name

RussellsParadox

Date of creation

2013-03-22 11:47:49

Last modified on

2013-03-22 11:47:49

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03-00

Synonym

axiom of unrestricted comprehension

Related topic

ZermeloFraenkelAxioms

Related topic

LambdaCalculus

Defines

axiom of comprehension

Generated on Thu Feb 8 19:20:07 2018 by

LaTeXML
set closed under an operation

set closed under an operation

A set

X

is said to be

closed

under some map

L

, if

L

maps

elements

in

X

to elements in

X

, i.e.,

L

:

X

→

X

.
More generally, suppose

Y

is the

n

-fold

Cartesian product

Y

=

X

×

⋯

×

X

. If

L

is a map

L

:

Y

→

X

, then we
also say that

X

is

closed under

the map

L

.

The above definition has no

relation

with the definition of a

closed set

in

topology

. Instead,
one should think of

X

and

L

as a closed
system.

Examples

1.

The set of

invertible matrices

is closed under

matrix inversion

.
This means that the

inverse

of an invertible matrix is again an
invertible matrix.

2.

Let

C

⁢

(

X

)

be the set of complex valued

continuous functions

on
some

topological space

X

.
Suppose

f

,

g

are

functions

in

C

⁢

(

X

)

. Then we define the
pointwise product of

f

and

g

as the function

f

⁢

g

:

x

↦

f

⁢

(

x

)

⁢

g

⁢

(

x

)

.
Since

f

⁢

g

is

continuous

, we have that

C

⁢

(

X

)

is closed under

pointwise

multiplication

.

In the first example, the

operation

is of the type

X

→

X

. In the latter,
pointwise multiplication is a map

C

⁢

(

X

)

×

C

⁢

(

X

)

→

C

⁢

(

X

)

.

The second example illustrated the somewhat odd definition of this term. When a function is defined, its domain and codomain are part of its definition, so it’s a little odd to talk about whether the function is closed or not: if you know what the function is, then you should know whether or not it is closed. So in practice, the way the term is used is this: We have a set

X

and we have a function

f

:

X

n

→

X

. We are given a subset

Y

, and asked whether

Y

is closed under

f

. In other words, there is a natural way to make

f

into a function

f

:

Y

n

→

X

, by

restriction

; the question is, does the result always lie in

Y

? This would mean that

f

yields a function

Y

n

→

Y

, which is usually what we want.

Occasionally the word is used in a potentially confusing way. For example,

left ideals

in a ring are supposed to be closed under addition (which is an example of what we just discussed) and left multiplication by arbitrary ring elements. What this last condition means is that for every

r

in the ring, the left ideal should be closed under the function

x

↦

r

⁢

x

.

Title

set closed under an operation

Canonical name

SetClosedUnderAnOperation

Date of creation

2013-03-22 13:49:49

Last modified on

2013-03-22 13:49:49

Owner

archibal (4430)

Last modified by

archibal (4430)

Numerical id

8

Author

archibal (4430)

Entry type

Definition

Classification

msc 03-00

Generated on Thu Feb 8 19:20:09 2018 by

LaTeXML
signature of a permutation

signature of a permutation

Let

X

be a

finite set

, and let

G

be the group of

permutations

of

X

(see

permutation group

). There exists a unique

homomorphism

χ

from

G

to the

multiplicative group

{

-

1

,

1

}

such that

χ

⁢

(

t

)

=

-

1

for any

transposition

(loc. sit.)

t

∈

G

. The value

χ

⁢

(

g

)

, for any

g

∈

G

, is called the

signature

or

sign

of the permutation

g

. If

χ

⁢

(

g

)

=

1

,

g

is said to be of even

parity

; if

χ

⁢

(

g

)

=

-

1

,

g

is said to be of odd parity.

Proposition

:

If

X

is

totally ordered

by
a

relation

<

, then for all

g

∈

G

,

χ

⁢

(

g

)

=

(

-

1

)

k

⁢

(

g

)

(1)

where

k

⁢

(

g

)

is the number of pairs

(

x

,

y

)

∈

X

×

X

such that

x

<

y

and

g

⁢

(

x

)

>

g

⁢

(

y

)

. (Such a pair is sometimes called an

inversion

of the permutation

g

.)

Proof:

This is clear if

g

is the

identity map

X

→

X

.
If

g

is any other permutation, then for some

consecutive

a

,

b

∈

X

we have

a

<

b

and

g

⁢

(

a

)

>

g

⁢

(

b

)

. Let

h

∈

G

be the transposition of

a

and

b

. We have

k

⁢

(

g

∘

h

)

=

k

⁢

(

g

)

-

1

χ

⁢

(

g

∘

h

)

=

-

χ

⁢

(

g

)

and the proposition follows by

induction

on

k

⁢

(

g

)

.

Title

signature of a permutation

Canonical

name

SignatureOfAPermutation

Date of creation

2013-03-22 13:29:19

Last modified on

2013-03-22 13:29:19

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

9

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 03-00

Classification

msc 05A05

Classification

msc 20B99

Synonym

sign of a permutation

Related topic

Transposition

Defines

inversion

Defines

signature

Defines

parity

Defines

even permutation

Defines

odd permutation

Generated on Thu Feb 8 19:20:12 2018 by

LaTeXML
simple function

simple function

In

measure theory

, a

simple function

is a

function

that is a
finite

linear combination

h

=

∑

k

=

1

n

c

k

⁢

χ

A

k

of

characteristic functions

, where the

c

k

are real

coefficients

and
every

A

k

is a

measurable set

with respect to a fixed

measure space

.

If the measure space is

ℝ

and each

A

k

is an

interval

,
then the function is called a

step function

. Thus, every step
function is a simple function.

Simple functions are used in

analysis

to interpolate between
characteristic functions and

measurable functions

. In other words,
characteristic functions are easy to

integrate

:

∫

E

χ

A

⁢

𝑑

x

=

|

A

|

,

while simple functions are not much harder to integrate:

∫

E

∑

k

=

1

n

c

k

⁢

χ

A

k

⁢

d

⁢

x

=

∑

k

=

1

n

c

k

⁢

|

A

k

|

.

To integrate a measurable function, one approximates it from below by
simple functions. Thus, simple functions can be used to define the

Lebesgue integral

over a subset of the measure space.

Title

simple function

Canonical

name

SimpleFunction

Date of creation

2013-03-22 12:21:16

Last modified on

2013-03-22 12:21:16

Owner

mps (409)

Last modified by

mps (409)

Numerical id

9

Author

mps (409)

Entry type

Definition

Classification

msc 03-00

Classification

msc 26A09

Classification

msc 26-00

Classification

msc 28-00

Related topic

CharacteristicFunction

Related topic

Integral2

Defines

step function

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

LaTeXML
subset

subset

Given two sets

A

and

B

, we say that

A

is a subset of

B

(which we denote as

A

⊆

B

or simply

A

⊂

B

) if every element of

A

is also in

B

. That is, the following

implication

holds:

x

∈

A

⇒

x

∈

B

.

The

relation

between

A

and

B

is then called

set inclusion

.

Some examples:

The set

A

=

{

d

,

r

,

i

,

t

,

o

}

is a subset of the set

B

=

{

p

,

e

,

d

,

r

,

i

,

t

,

o

}

because every element of

A

is also in

B

. That is,

A

⊆

B

.

On the other hand, if

C

=

{

p

,

e

,

d

,

r

,

o

}

, then neither

A

⊆

C

(because

t

∈

A

but

t

∉

C

) nor

C

⊆

A

(because

p

∈

C

but

p

∉

A

). The fact that

A

is not a subset of

C

is written as

A

⊈

C

. Similarly, we have

C

⊈

A

.

If

X

⊆

Y

and

Y

⊆

X

, it must be the case that

X

=

Y

.

Every set is a subset of itself, and the

empty set

is a subset of every other set. The set

A

is called a

proper subset

of

B

, if

A

⊂

B

and

A

≠

B

. In this case, we do not use

A

⊆

B

.

Title

subset

Canonical

name

Subset

Date of creation

2013-03-22 11:52:38

Last modified on

2013-03-22 11:52:38

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

13

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 03-00

Classification

msc 00-02

Related topic

EmptySet

Related topic

Superset

Related topic

TotallyBounded

Related topic

ProofThatAllSubgroupsOfACyclicGroupAreCyclic

Related topic

Property2

Related topic

CardinalityOfAFiniteSetIsUnique

Related topic

CriterionOfSurjectivity

Defines

set inclusion

Generated on Thu Feb 8 19:20:16 2018 by

LaTeXML
surjective

surjective

A

function

f

:

X

→

Y

is called

surjective

or

onto

if, for every

y

∈

Y

, there is an

x

∈

X

such that

f

⁢

(

x

)

=

y

.

Equivalently,

f

:

X

→

Y

is onto when its image is all the codomain:

Im

⁢

f

=

Y

.

Properties

1.

If

f

:

X

→

Y

is any function, then

f

:

X

→

f

⁢

(

X

)

is
a surjection. That is, by restricting the codomain,
any function induces a surjection.

2.

The

composition

of surjective functions (when defined) is
again a surjective function.

3.

If

f

:

X

→

Y

is a surjection and

B

⊆

Y

, then
(see this page (

http://planetmath.org/InverseImage

))

f

⁢

f

-

1

⁢

(

B

)

=

B

.

Title

surjective

Canonical

name

Surjective

Date of creation

2013-03-22 12:32:48

Last modified on

2013-03-22 12:32:48

Owner

drini (3)

Last modified by

drini (3)

Numerical id

7

Author

drini (3)

Entry type

Definition

Classification

msc 03-00

Synonym

onto

Related topic

TypesOfHomomorphisms

Related topic

InjectiveFunction

Related topic

Bijection

Related topic

Function

Related topic

OneToOneFunctionFromOntoFunction

Defines

surjection

Generated on Thu Feb 8 19:20:18 2018 by

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topic entry on foundations of mathematics

topic entry on foundations of mathematics

This is an entry-in-progress for a new topic entry on the

foundations of mathematics

.

1.

Syntax and

semantics

2.

Set theory

and Classes

3.

Concepts in set theory

4.

Index of set theory

5.

Logic and Axiomatic theories

6.

Category theory

7.

Model theory

8.

Proof theory

: from axioms to results

9.

Symbolic computation

10.

Theory of computation

Title

topic entry on foundations of mathematics

Canonical name

TopicEntryOnFoundationsOfMathematics

Date of creation

2013-03-22 18:00:10

Last modified on

2013-03-22 18:00:10

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

32

Author

CWoo (3771)

Entry type

Topic

Classification

msc 03-00

Related topic

QuantumLogic

Generated on Thu Feb 8 19:20:27 2018 by

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transposition

transposition

Given a

finite set

X

=

{

a

1

,

a

2

,

…

,

a

n

}

, a

transposition

is a

permutation

(

bijective function

of

X

onto itself)

f

such that there exist indices

i

,

j

such that

f

⁢

(

a

i

)

=

a

j

,

f

⁢

(

a

j

)

=

a

i

and

f

⁢

(

a

k

)

=

a

k

for all other indices

k

. This is often denoted (in the

cycle notation

) as

(

a

,

b

)

.

Example:
If

X

=

{

a

,

b

,

c

,

d

,

e

}

the

function

σ

given by

σ

⁢

(

a

)

=

a

σ

⁢

(

b

)

=

e

σ

⁢

(

c

)

=

c

σ

⁢

(

d

)

=

d

σ

⁢

(

e

)

=

b

is a transposition.

One of the main results on

symmetric groups

states that any permutation can be expressed as

composition

(

product

) of transpositions, and for any two

decompositions

of a given permutation, the number of transpositions is always even or always odd.

Title

transposition

Canonical name

Transposition

Date of creation

2013-03-22 12:24:30

Last modified on

2013-03-22 12:24:30

Owner

drini (3)

Last modified by

drini (3)

Numerical id

6

Author

drini (3)

Entry type

Definition

Classification

msc 03-00

Classification

msc 05A05

Classification

msc 20B99

Related topic

Cycle2

Related topic

SignatureOfAPermutation

Generated on Thu Feb 8 19:20:29 2018 by

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truth table

truth table

A

truth table

is a tabular listing of all possible input value

combinations

for a logical

function

and their corresponding output values. Similarly, the truth table of a logical

proposition

is the truth table of the corresponding logical function.

For

instance

, the truth table of the

connective

“or” is as follows:

a

b

a

∨

b

F

F

F

F

T

T

T

F

T

T

T

T

For

n

input

variables

, there will always be

2

n

rows in the truth table.
A sample truth table for “

(

a

∧

b

)

→

c

” would be

a

b

c

(

a

∧

b

)

→

c

F

F

F

T

F

F

T

F

F

T

F

T

F

T

T

F

T

F

F

T

T

F

T

F

T

T

F

T

T

T

T

T

(Note that

∧

represents

logical and

, while

→

represents the

conditional truth function

).

To compute truth tables of

expressions

, one often proceeds in steps. for instance,
to compute a truth table for “

¬

⁢

p

∨

(

p

∧

q

)

, one might proceed as follows:

p

q

¬

⁢

p

(

p

∧

q

)

¬

⁢

p

∨

(

p

∧

q

)

F

F

T

F

T

F

T

T

F

T

T

F

F

F

F

T

T

F

T

T

For reference, here is a truth table of some popular connectives:

p

q

p

∨

q

p

∧

q

p

⊻

q

p

→

q

p

↔

q

F

F

F

F

F

T

T

F

T

T

F

T

T

F

T

F

T

F

T

F

F

T

T

T

T

F

T

T

For completeness, here are the remaining connectives, excluding

trivial

connectives which
depend on only one or none of their

arguments

:

p

q

p

⁢

∧

q

p

⁢

∨

q

p

←

q

p

↛

q

p

⁢

←

q

F

F

T

T

T

F

F

F

T

T

F

F

F

T

T

F

T

F

T

T

F

T

T

F

F

T

F

F

Title

truth table

Canonical

name

TruthTable

Date of creation

2013-03-22 11:54:35

Last modified on

2013-03-22 11:54:35

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

16

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 03-00

Classification

msc 34C29

Related topic

ZerothOrderLogic

Related topic

PropositionalCalculus

Generated on Thu Feb 8 19:20:32 2018 by

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(a,b)=(c,d) if and only if a=c and b=d

(

a

,

b

)

=

(

c

,

d

)

if and only if

a

=

c

and

b

=

d

Following is a proof that the

ordered pairs

(

a

,

b

)

and

(

c

,

d

)

are equal if and only if

a

=

c

and

b

=

d

.

Proof.

If

a

=

c

and

b

=

d

, then

(

a

,

b

)

=

{

{

a

}

,

{

a

,

b

}

}

=

{

{

c

}

,

{

c

,

d

}

}

=

(

c

,

d

)

.

Assume that

(

a

,

b

)

=

(

c

,

d

)

and

a

=

b

. Then

{

{

c

}

,

{

c

,

d

}

}

=

(

c

,

d

)

=

(

a

,

b

)

=

{

{

a

}

,

{

a

,

b

}

}

=

{

{

a

}

,

{

a

,

a

}

}

=

{

{

a

}

,

{

a

}

}

=

{

{

a

}

}

.
Thus,

{

c

,

d

}

∈

{

{

a

}

}

. Therefore,

{

c

,

d

}

=

{

a

}

. Hence,

a

=

c

and

a

=

d

. Since it was also assumed that

a

=

b

, it follows that

a

=

c

and

b

=

d

.

Finally, assume that

(

a

,

b

)

=

(

c

,

d

)

and

a

≠

b

. Then

{

a

}

≠

{

a

,

b

}

. Note that

{

{

a

}

,

{

a

,

b

}

}

=

(

a

,

b

)

=

(

c

,

d

)

=

{

{

c

}

,

{

c

,

d

}

}

. Thus,

{

c

}

∈

{

{

a

}

,

{

a

,

b

}

}

. It cannot be the case that

{

c

}

=

{

a

,

b

}

(lest

a

=

c

=

b

). Thus,

{

c

}

=

{

a

}

. Therefore,

a

=

c

. Hence,

{

{

a

}

,

{

a

,

b

}

}

=

{

{

c

}

,

{

c

,

d

}

}

=

{

{

a

}

,

{

a

,

d

}

}

. Note that

{

a

,

b

}

∈

{

{

a

}

,

{

a

,

d

}

}

. Since

{

a

}

≠

{

a

,

b

}

, it must be the case that

{

a

,

b

}

=

{

a

,

d

}

. Thus,

b

∈

{

a

,

d

}

. Since

a

≠

b

, it must be the case that

b

=

d

. It follows that

a

=

c

and

b

=

d

.
∎

Title

(

a

,

b

)

=

(

c

,

d

)

if and only if

a

=

c

and

b

=

d

Canonical name

abcdIfAndOnlyIfAcAndBd

Date of creation

2013-03-22 16:13:19

Last modified on

2013-03-22 16:13:19

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

9

Author

Wkbj79 (1863)

Entry type

Proof

Classification

msc 03-00

Generated on Thu Feb 8 19:20:36 2018 by

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paradox

paradox

A

paradox

is an assertion that is apparently self-contradictory, though based on a valid

deduction

from acceptable

premises

.

Paradoxes typically lead to a reevaluation of the axioms of mathematics. after axioms are assumed so that the paradox is averted, the statement is still usually referred to as a paradox.

Occasionally, one may refer to a surprising result as a paradox. Such is the case in the

birthday paradox

, which is not apparently self-contradictory.

Examples of paradoxes include:

•

Banach-Tarski paradox

•

binary tree paradox

•

birthday paradox (

http://planetmath.org/BirthdayParadox

)

•

Burali-Forti paradox

•

Cantor’s paradox

•

Hausdorff paradox

•

Russell’s paradox

•

Simpson’s paradox

•

Zeno’s paradox

Title

paradox

Canonical name

Paradox

Date of creation

2013-03-22 16:07:49

Last modified on

2013-03-22 16:07:49

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

14

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 03A05

Classification

msc 03B99

Synonym

paradoxical

Synonym

paradoxically

Synonym

dilemma

Related topic

GalileosParadox

Generated on Thu Feb 8 19:21:30 2018 by

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