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Sep 21, 2022, 1:22:18 AM9/21/22

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Here is the truth table for the IMPLIES-operator:

A B A=>B

T T T

T F F

F T T

F F T

Judging by frequent questions at MSE (at least once every day) on this topic, there seems to be much confusion about this one, simple table.

This table is often used to "define" the IMPLIES-operator in introductory textbooks. No real explanation is usually given, just a few examples. There is a good reason for this, namely that to truly understand why it works, you must understand some basic methods of proof, i.e. the following rules of inference in propositional logic:

1. Premise

2. Conclusion (intro =>, intro ~)

3. Join (intro &)

4. Split (elim &)

5. Detachment (elim =>)

6. Remove ~~

(For an excellent introduction to these rules of inference, you need only work your way through only the first 3 (of 13) examples in the DC Proof tutorial.)

The above truth table is really just a table of the following 4 theorems in propositional logic:

1. A & B => [A => B]

2. A & ~B => ~[A => B]

3. ~A & B => [A => B]

4. ~A & ~B => [A => B]

*****************************************************************

Thm 1: A & B => [A => B]

1. A & B

Premise

2. A

Premise

3. B

Split, 1

4. A => B

Conclusion, 2

5. A & B => [A => B]

Conclusion, 1

*****************************************************************

Thm 2: A & ~B => ~[A => B]

1. A & ~B

Premise

2. A => B

Premise

3. A

Split, 1

4. ~B

Split, 1

5. B

Detach, 2, 3

6. ~B & B

Join, 4, 5

7. ~[A => B]

Conclusion, 2

8. A & ~B => ~[A => B]

Conclusion, 1

*****************************************************************

Thm 3: ~A & B => [A => B]

1. ~A & B

Premise

2. A

Premise

3. B

Split, 1

4. A => B

Conclusion, 2

5. ~A & B => [A => B]

Conclusion, 1

*****************************************************************

Thm 4: ~A & ~B => [A => B]

1. ~A & ~B

Premise

2. A

Premise

3. ~B

Premise

4. ~A

Split, 1

5. A & ~A

Join, 2, 4

6. ~~B

Conclusion, 3

7. B

Rem DNeg, 6

8. A => B

Conclusion, 2

9. ~A & ~B => [A => B]

Conclusion, 1

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com

Visit my Math Blog at http://www.dcproof.wordpress.com

A B A=>B

T T T

T F F

F T T

F F T

Judging by frequent questions at MSE (at least once every day) on this topic, there seems to be much confusion about this one, simple table.

This table is often used to "define" the IMPLIES-operator in introductory textbooks. No real explanation is usually given, just a few examples. There is a good reason for this, namely that to truly understand why it works, you must understand some basic methods of proof, i.e. the following rules of inference in propositional logic:

1. Premise

2. Conclusion (intro =>, intro ~)

3. Join (intro &)

4. Split (elim &)

5. Detachment (elim =>)

6. Remove ~~

(For an excellent introduction to these rules of inference, you need only work your way through only the first 3 (of 13) examples in the DC Proof tutorial.)

The above truth table is really just a table of the following 4 theorems in propositional logic:

1. A & B => [A => B]

2. A & ~B => ~[A => B]

3. ~A & B => [A => B]

4. ~A & ~B => [A => B]

*****************************************************************

Thm 1: A & B => [A => B]

1. A & B

Premise

2. A

Premise

3. B

Split, 1

4. A => B

Conclusion, 2

5. A & B => [A => B]

Conclusion, 1

*****************************************************************

Thm 2: A & ~B => ~[A => B]

1. A & ~B

Premise

2. A => B

Premise

3. A

Split, 1

4. ~B

Split, 1

5. B

Detach, 2, 3

6. ~B & B

Join, 4, 5

7. ~[A => B]

Conclusion, 2

8. A & ~B => ~[A => B]

Conclusion, 1

*****************************************************************

Thm 3: ~A & B => [A => B]

1. ~A & B

Premise

2. A

Premise

3. B

Split, 1

4. A => B

Conclusion, 2

5. ~A & B => [A => B]

Conclusion, 1

*****************************************************************

Thm 4: ~A & ~B => [A => B]

1. ~A & ~B

Premise

2. A

Premise

3. ~B

Premise

4. ~A

Split, 1

5. A & ~A

Join, 2, 4

6. ~~B

Conclusion, 3

7. B

Rem DNeg, 6

8. A => B

Conclusion, 2

9. ~A & ~B => [A => B]

Conclusion, 1

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com

Visit my Math Blog at http://www.dcproof.wordpress.com

Sep 21, 2022, 3:04:48 AM9/21/22

to

If you know how material implications works, why do you write

nonsense like this here:

> > y=\=x => ~Bijection(id_y,x,x)

> Not true in general. What is your point, Jan Burse?

Do you have a pair y,x where the above statement

y=\=x => ~Bijection(id_y,x,x) is not true?

Hint: Must be a statement of the second row:

A B A=>B

T F F

nonsense like this here:

> > y=\=x => ~Bijection(id_y,x,x)

> Not true in general. What is your point, Jan Burse?

Do you have a pair y,x where the above statement

y=\=x => ~Bijection(id_y,x,x) is not true?

Hint: Must be a statement of the second row:

A B A=>B

T F F

Sep 21, 2022, 9:56:33 AM9/21/22

to

> If you know how material implications works, why do you write

> nonsense like this here:

>

> > > y=\=x => ~Bijection(id_y,x,x)

> > Not true in general. What is your point, Jan Burse?

>

> Do you have a pair y,x where the above statement

> y=\=x => ~Bijection(id_y,x,x) is not true?

>

Suppose x = {0} and y = {0, 1}. Then x=/=y and for all z in x, id_y(z) =z, i.e. id_y is a bijection on set x.
> nonsense like this here:

>

> > > y=\=x => ~Bijection(id_y,x,x)

> > Not true in general. What is your point, Jan Burse?

>

> Do you have a pair y,x where the above statement

> y=\=x => ~Bijection(id_y,x,x) is not true?

>

Sep 21, 2022, 11:47:39 AM9/21/22

to

Message has been deleted

Sep 21, 2022, 12:08:55 PM9/21/22

to

On Wednesday, September 21, 2022 at 5:47:39 PM UTC+2, Ross A. Finlayson wrote:

> What's 0^0 again?

1 .

By two different definitions:

a.) ^ defined recursively on IN:

Let m e IN:

m^0 = 1

m^(n+1) = m^n * m (for all n e IN)

b.) "set-theoretically" on IN (where IN is defined due to von Neumann):

m^n = card({f : n --> m}) (for all n, m e IN)

> What's 0^0 again?

1 .

By two different definitions:

a.) ^ defined recursively on IN:

Let m e IN:

m^0 = 1

m^(n+1) = m^n * m (for all n e IN)

b.) "set-theoretically" on IN (where IN is defined due to von Neumann):

m^n = card({f : n --> m}) (for all n, m e IN)

Sep 21, 2022, 1:28:57 PM9/21/22

to

On Wednesday, September 21, 2022 at 1:22:18 AM UTC-4, Dan Christensen wrote:

> Here is the truth table for the IMPLIES-operator:

>

> A B A=>B

> T T T

> T F F

> F T T

> F F T

>

> Judging by frequent questions at MSE (at least once every day) on this topic, there seems to be much confusion about this one, simple table.

>

> This table is often used to "define" the IMPLIES-operator in introductory textbooks. No real explanation is usually given, just a few examples. There is a good reason for this, namely that to truly understand why it works, you must understand some basic methods of proof, i.e. the following rules of inference in propositional logic:

>

> 1. Premise

> 2. Conclusion (intro =>, intro ~)

> 3. Join (intro &)

> 4. Split (elim &)

> 5. Detachment (elim =>)

> 6. Remove ~~

>

> (For an excellent introduction to these rules of inference, you need only work your way through only the first 3 (of 13) examples in the DC Proof tutorial.)

>

> The above truth table is really just a table of the following 4 theorems in propositional logic:

>

> 1. A & B => [A => B]

>

> 2. A & ~B => ~[A => B]

>

> 3. ~A & B => [A => B]

>

> 4. ~A & ~B => [A => B]

>

Started working on an update to the DC Proof tutorial to include these theorems as exercises with hints and full solutions. Should have thought of this years ago! The perfect proofs for the beginner--no more than 9 lines each and you don't even need that much logic. And IMHO quite profound when taken as a whole.
> Here is the truth table for the IMPLIES-operator:

>

> A B A=>B

> T T T

> T F F

> F T T

> F F T

>

> Judging by frequent questions at MSE (at least once every day) on this topic, there seems to be much confusion about this one, simple table.

>

> This table is often used to "define" the IMPLIES-operator in introductory textbooks. No real explanation is usually given, just a few examples. There is a good reason for this, namely that to truly understand why it works, you must understand some basic methods of proof, i.e. the following rules of inference in propositional logic:

>

> 1. Premise

> 2. Conclusion (intro =>, intro ~)

> 3. Join (intro &)

> 4. Split (elim &)

> 5. Detachment (elim =>)

> 6. Remove ~~

>

> (For an excellent introduction to these rules of inference, you need only work your way through only the first 3 (of 13) examples in the DC Proof tutorial.)

>

> The above truth table is really just a table of the following 4 theorems in propositional logic:

>

> 1. A & B => [A => B]

>

> 2. A & ~B => ~[A => B]

>

> 3. ~A & B => [A => B]

>

> 4. ~A & ~B => [A => B]

>

Sep 21, 2022, 2:01:08 PM9/21/22

to

On Wednesday, September 21, 2022 at 11:47:39 AM UTC-4, Ross A. Finlayson wrote:

[snip]

> What's 0^0 again?

Huh? Oh, I see... You would rather change the subject. Can't blame you, really.

See: https://dcproof.wordpress.com/2013/10/09/oh-the-ambiguity-2/

Start a new thread if you want to rehash it after all these years. Could be fun.

[snip]

> What's 0^0 again?

Huh? Oh, I see... You would rather change the subject. Can't blame you, really.

See: https://dcproof.wordpress.com/2013/10/09/oh-the-ambiguity-2/

Start a new thread if you want to rehash it after all these years. Could be fun.

Sep 21, 2022, 7:55:31 PM9/21/22

to

On Wednesday, September 21, 2022 at 1:28:57 PM UTC-4, Dan Christensen wrote:

> On Wednesday, September 21, 2022 at 1:22:18 AM UTC-4, Dan Christensen wrote:

> > Here is the truth table for the IMPLIES-operator:

> >

> > A B A=>B

> > T T T

> > T F F

> > F T T

> > F F T

> >

> > Judging by frequent questions at MSE (at least once every day) on this topic, there seems to be much confusion about this one, simple table.

> >

> > This table is often used to "define" the IMPLIES-operator in introductory textbooks. No real explanation is usually given, just a few examples. There is a good reason for this, namely that to truly understand why it works, you must understand some basic methods of proof, i.e. the following rules of inference in propositional logic:

> >

> > 1. Premise

> > 2. Conclusion (intro =>, intro ~)

> > 3. Join (intro &)

> > 4. Split (elim &)

> > 5. Detachment (elim =>)

> > 6. Remove ~~

> >

> > (For an excellent introduction to these rules of inference, you need only work your way through only the first 3 (of 13) examples in the DC Proof tutorial.)

> >

> > The above truth table is really just a table of the following 4 theorems in propositional logic:

> >

> > 1. A & B => [A => B]

> >

> > 2. A & ~B => ~[A => B]

> >

> > 3. ~A & B => [A => B]

> >

> > 4. ~A & ~B => [A => B]

> >

> Started working on an update to the DC Proof tutorial to include these theorems as exercises with hints and full solutions. Should have thought of this years ago! The perfect proofs for the beginner--no more than 9 lines each and you don't even need that much logic. And IMHO quite profound when taken as a whole.

Now available at my homepage.
> On Wednesday, September 21, 2022 at 1:22:18 AM UTC-4, Dan Christensen wrote:

> > Here is the truth table for the IMPLIES-operator:

> >

> > A B A=>B

> > T T T

> > T F F

> > F T T

> > F F T

> >

> > Judging by frequent questions at MSE (at least once every day) on this topic, there seems to be much confusion about this one, simple table.

> >

> > This table is often used to "define" the IMPLIES-operator in introductory textbooks. No real explanation is usually given, just a few examples. There is a good reason for this, namely that to truly understand why it works, you must understand some basic methods of proof, i.e. the following rules of inference in propositional logic:

> >

> > 1. Premise

> > 2. Conclusion (intro =>, intro ~)

> > 3. Join (intro &)

> > 4. Split (elim &)

> > 5. Detachment (elim =>)

> > 6. Remove ~~

> >

> > (For an excellent introduction to these rules of inference, you need only work your way through only the first 3 (of 13) examples in the DC Proof tutorial.)

> >

> > The above truth table is really just a table of the following 4 theorems in propositional logic:

> >

> > 1. A & B => [A => B]

> >

> > 2. A & ~B => ~[A => B]

> >

> > 3. ~A & B => [A => B]

> >

> > 4. ~A & ~B => [A => B]

> >

> Started working on an update to the DC Proof tutorial to include these theorems as exercises with hints and full solutions. Should have thought of this years ago! The perfect proofs for the beginner--no more than 9 lines each and you don't even need that much logic. And IMHO quite profound when taken as a whole.

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com <------- Here

Sep 21, 2022, 8:00:20 PM9/21/22

to

Let n, m e IN. Then

n + m = card(n x {0} u m x {1}) ,

n * m = card(n x m) ,

n ^ m = card({f | f : m --> n}) .

n = 0 and/or m = 0 is no "special case" here.

Actually, we would "artificially" have to treat 0^0 as a special case to "avoid" the result 1. But there is absolutely no reason to do so (in this context).

Sep 21, 2022, 9:54:01 PM9/21/22

to

just pointing out that your stipulations and

"showing your work" about fail given alternatives.

I know you "have to have it your way", but when it comes

to usual set-theoretic disambiguation, you're just a usual

shill and fool.

Showing that now your "implies is inference" also fails

your other failings about the maintenance of the truth-valued

in truth tables and no "false antecedents",

you false antecedent.

Just change the order of stipulations of your proofs they

change what they say. Don't you think inference should

be a little more reliable than assuming what you set out to prove?

No offense, it's a little "show your work" tool for some reasoning,

but it's weak and compounded by "false antecedents in a world

of truth tables", it's further "garbage-in garbage-out".

Then, really I do not care what you think "0^0" is,

just pointing out that you think you do,

but there's more than one answer, so,

one of yours must be wrong.

Sep 21, 2022, 10:37:33 PM9/21/22

to

On Wednesday, September 21, 2022 at 9:54:01 PM UTC-4, Ross A. Finlayson wrote:

> On Wednesday, September 21, 2022 at 11:01:08 AM UTC-7, Dan Christensen wrote:

> > On Wednesday, September 21, 2022 at 11:47:39 AM UTC-4, Ross A. Finlayson wrote:

> >

> > [snip]

> >

> > > What's 0^0 again?

> >

> > Huh? Oh, I see... You would rather change the subject. Can't blame you, really.

> >

> > See: https://dcproof.wordpress.com/2013/10/09/oh-the-ambiguity-2/

> >

> > Start a new thread if you want to rehash it after all these years. Could be fun.

[snip]
> On Wednesday, September 21, 2022 at 11:01:08 AM UTC-7, Dan Christensen wrote:

> > On Wednesday, September 21, 2022 at 11:47:39 AM UTC-4, Ross A. Finlayson wrote:

> >

> > [snip]

> >

> > > What's 0^0 again?

> >

> > Huh? Oh, I see... You would rather change the subject. Can't blame you, really.

> >

> > See: https://dcproof.wordpress.com/2013/10/09/oh-the-ambiguity-2/

> >

> > Start a new thread if you want to rehash it after all these years. Could be fun.

So, you would rather not. Very wise.

> Don't you think inference should

> be a little more reliable than assuming what you set out to prove?

>

Where do you imagine that I assume ~A & B => [A => B] ?

Here again is my proof:

1. ~A & B

Premise

2. A

Premise

3. B

Split, 1

4. A => B

Conclusion, 2

5. ~A & B => [A => B]

Conclusion, 1

Sep 22, 2022, 11:30:55 AM9/22/22

to

For example, A as "A is premise 1", i.e. whether A or not A,

then "A is premise 2", it's an example of a generic premise

that according to its introduction, and what so follows,

doesn't admit false antecedents, at all.

What otherwise you have there as a template for "wrong".

Sep 22, 2022, 11:59:57 AM9/22/22

to

all these guns that shoot backward.

Then you forget which ones you put where, and none of them are safe.

You got no reliable reverse "material" implication.

What you've written there is "not-A not-implies-B", not, "not-A not not-implies-not-B",

that a false antecedent is always "not implies".

Which is not necessarily "implies".

Leave "material" "implication" out of things, direct implication only. It's about

one of the stupidest kinds of notational laziness.

Don't you have any kind of object in your theory that changes?

Can't a _change_ even possibly be an object in your little theory?

Then it's like you say "whu all these are constants" then it's like

"you mean your variables, ...".

Sep 22, 2022, 12:42:51 PM9/22/22

to

Nope, id_y isn't a bijection x -> x, it has missing

dom(id_y) = x

id_y is not member of this function space:

{ f : x -> x | f bijective }

Whats wrong with you?

dom(id_y) = x

id_y is not member of this function space:

{ f : x -> x | f bijective }

Whats wrong with you?

Sep 22, 2022, 12:47:29 PM9/22/22

to

On Thursday, September 22, 2022 at 11:30:55 AM UTC-4, Ross A. Finlayson wrote:

> On Wednesday, September 21, 2022 at 7:37:33 PM UTC-7, Dan Christensen wrote:

> > On Wednesday, September 21, 2022 at 9:54:01 PM UTC-4, Ross A. Finlayson wrote:

> > > On Wednesday, September 21, 2022 at 11:01:08 AM UTC-7, Dan Christensen wrote:

> > > > On Wednesday, September 21, 2022 at 11:47:39 AM UTC-4, Ross A. Finlayson wrote:

> > > >

> > > > [snip]

> > > >

> > > > > What's 0^0 again?

> > > >

> > > > Huh? Oh, I see... You would rather change the subject. Can't blame you, really.

> > > >

> > > > See: https://dcproof.wordpress.com/2013/10/09/oh-the-ambiguity-2/

> > > >

> > > > Start a new thread if you want to rehash it after all these years. Could be fun.

> > [snip]

> >

> > So, you would rather not. Very wise.

> > > Don't you think inference should

> > > be a little more reliable than assuming what you set out to prove?

> > >

> > [snip]

> >

> > Where do you imagine that I assume ~A & B => [A => B] ?

> >

No comment?
> On Wednesday, September 21, 2022 at 7:37:33 PM UTC-7, Dan Christensen wrote:

> > On Wednesday, September 21, 2022 at 9:54:01 PM UTC-4, Ross A. Finlayson wrote:

> > > On Wednesday, September 21, 2022 at 11:01:08 AM UTC-7, Dan Christensen wrote:

> > > > On Wednesday, September 21, 2022 at 11:47:39 AM UTC-4, Ross A. Finlayson wrote:

> > > >

> > > > [snip]

> > > >

> > > > > What's 0^0 again?

> > > >

> > > > Huh? Oh, I see... You would rather change the subject. Can't blame you, really.

> > > >

> > > > See: https://dcproof.wordpress.com/2013/10/09/oh-the-ambiguity-2/

> > > >

> > > > Start a new thread if you want to rehash it after all these years. Could be fun.

> > [snip]

> >

> > So, you would rather not. Very wise.

> > > Don't you think inference should

> > > be a little more reliable than assuming what you set out to prove?

> > >

> > [snip]

> >

> > Where do you imagine that I assume ~A & B => [A => B] ?

> >

> > Here again is my proof:

> > 1. ~A & B

> > Premise

> >

> > 2. A

> > Premise

> >

> > 3. B

> > Split, 1

> >

> > 4. A => B

> > Conclusion, 2

> >

> > 5. ~A & B => [A => B]

> > Conclusion, 1

> You have no concept of time, the modal, or change.

>

I am looking the state of "the world" at some instant in time (usually the present). A snapshot. It may sound limiting, but it turns out to be extremely useful in the real world and in mathematics.
>

The statement, "If it is raining then it is cloudy" means only that, at present, it is not both raining and not cloudy. It does NOT mean that rain causes cloudiness. Or that, historically, whenever it was raining it was also cloudy.

> For example, A as "A is premise 1", i.e. whether A or not A,

> then "A is premise 2", it's an example of a generic premise

> that according to its introduction, and what so follows,

> doesn't admit false antecedents, at all.

>

EXAMPLE

1 A & ~A

Premise

2 ~[A & ~A]

Sep 22, 2022, 12:49:00 PM9/22/22

to

Usually in set theory this here should hold for product function spaces:

Definition: 4.2.6 The Exponentiation Axiom (abbreviated Exp) postulates

that for sets a; b the class of all functions from a to b forms a set:

∀a∀b∃c ∀f(f e c <-> (f : a -> b))

https://www1.maths.leeds.ac.uk/~rathjen/book.pdf

You can even strengthen it to ∃!c. Now it seems in

DC Spoiled, the notion of product function space is

shape shifting, and so are function spaces that are

subset of the product function space, like this here:

{ f : x -> x | f bijective }

If id_y is element of the above function space, this

would imply that x -> x and y -> y are not disjoint,

for x =\= y. But usually they are disjoint, even in the

set theoretic approach, and also in the Bourbaki approach.

LMAO!

Definition: 4.2.6 The Exponentiation Axiom (abbreviated Exp) postulates

that for sets a; b the class of all functions from a to b forms a set:

∀a∀b∃c ∀f(f e c <-> (f : a -> b))

https://www1.maths.leeds.ac.uk/~rathjen/book.pdf

You can even strengthen it to ∃!c. Now it seems in

DC Spoiled, the notion of product function space is

shape shifting, and so are function spaces that are

subset of the product function space, like this here:

{ f : x -> x | f bijective }

would imply that x -> x and y -> y are not disjoint,

for x =\= y. But usually they are disjoint, even in the

set theoretic approach, and also in the Bourbaki approach.

LMAO!

Sep 22, 2022, 1:13:28 PM9/22/22

to

> What you've written there is "not-A not-implies-B", not, "not-A not not-implies-not-B",

> that a false antecedent is always "not implies".

>

> Which is not necessarily "implies".

>

> Leave "material" "implication" out of things, direct implication only. It's about

> one of the stupidest kinds of notational laziness.

>

1. Premise

2. Conclusion (intro =>, intro ~)

3. Join (intro &)

4. Split (elim &)

5. Detachment (elim =>)

6. Remove ~~

> Don't you have any kind of object in your theory that changes?

You could interpret some independent variable t as time with different values of t corresponding to different points in space if that's what you mean? But that's getting into the realm of science.

Sep 22, 2022, 1:29:14 PM9/22/22

to

On Thursday, September 22, 2022 at 12:49:00 PM UTC-4, Mostowski Collapse wrote:

[snip]

> Usually in set theory this here should hold for product function spaces:

>

> Definition: 4.2.6 The Exponentiation Axiom (abbreviated Exp) postulates

> that for sets a; b the class of all functions from a to b forms a set:

> ∀a∀b∃c ∀f(f e c <-> (f : a -> b))

> https://www1.maths.leeds.ac.uk/~rathjen/book.pdf

>

Not unlike the DC Proof Function Space axiom:

ALL(dom):ALL(cod):[Set(dom) & Set(cod)

=> EXIST(fsp):[Set(fsp) & ALL(f):[f in fsp <=> Function(f,dom,cod) & ALL(a1):[a1 in dom => f(a1) in cod]]]]

> You can even strengthen it to ∃!c. Now it seems in

> DC Proof, the notion of product function space is

[snip]

> Usually in set theory this here should hold for product function spaces:

>

> Definition: 4.2.6 The Exponentiation Axiom (abbreviated Exp) postulates

> that for sets a; b the class of all functions from a to b forms a set:

> ∀a∀b∃c ∀f(f e c <-> (f : a -> b))

> https://www1.maths.leeds.ac.uk/~rathjen/book.pdf

>

ALL(dom):ALL(cod):[Set(dom) & Set(cod)

=> EXIST(fsp):[Set(fsp) & ALL(f):[f in fsp <=> Function(f,dom,cod) & ALL(a1):[a1 in dom => f(a1) in cod]]]]

> You can even strengthen it to ∃!c. Now it seems in

> shape shifting, and so are function spaces that are

> subset of the product function space, like this here:

> { f : x -> x | f bijective }

> If id_y is element of the above function space, this

> would imply that x -> x and y -> y are not disjoint,

> for x =\= y.

If, however, x is a proper subset of y, we can say that id_y (the identity function on set y) is also a bijection on set x .
> subset of the product function space, like this here:

> { f : x -> x | f bijective }

> If id_y is element of the above function space, this

> would imply that x -> x and y -> y are not disjoint,

> for x =\= y.

Sep 22, 2022, 4:06:24 PM9/22/22

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Dan Christensen halucinated on Thursday, 22. September 2022:

What does exist in mathematics, is the operation restriction:

f | A = { (x,y) e f | x e A }

And you can for example say:

id_X = id_Y | X

for X ⊆ Y. And subsequetly you can also say id_Y | X is a X -> X bijection.

This is all explained in the very first chapters here:

Basic Set Theory (Dover Books on Mathematics)

Azriel Levy - Revised Edition (13. August 2002)

https://www.amazon.com/dp/0486420795

Or read wikipedia:

In mathematics, the restriction of a function f is a new function, denoted f | A

https://en.wikipedia.org/wiki/Restriction_%28mathematics%29

> If, however, x is a proper subset of y, we can say that id_y (the identity function on set y) is also a bijection on set x .

No you cannot say that. People don't say that.
What does exist in mathematics, is the operation restriction:

f | A = { (x,y) e f | x e A }

And you can for example say:

id_X = id_Y | X

for X ⊆ Y. And subsequetly you can also say id_Y | X is a X -> X bijection.

This is all explained in the very first chapters here:

Basic Set Theory (Dover Books on Mathematics)

Azriel Levy - Revised Edition (13. August 2002)

https://www.amazon.com/dp/0486420795

Or read wikipedia:

In mathematics, the restriction of a function f is a new function, denoted f | A

https://en.wikipedia.org/wiki/Restriction_%28mathematics%29

Sep 22, 2022, 5:09:31 PM9/22/22

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On Thursday, September 22, 2022 at 4:06:24 PM UTC-4, Mostowski Collapse wrote:

> > If, however, x is a proper subset of y, we can say that id_y (the identity function on set y) is also a bijection on set x .

> No you cannot say that. People don't say that.

>

> What does exist in mathematics, is the operation restriction:

>

> f | A = { (x,y) e f | x e A }

>

> And you can for example say:

>

> id_X = id_Y | X

>

> for X ⊆ Y. And subsequetly you can also say id_Y | X is a X -> X bijection.

>

In other words, like I said, "id_y is a bijection on set x."
> > If, however, x is a proper subset of y, we can say that id_y (the identity function on set y) is also a bijection on set x .

> No you cannot say that. People don't say that.

>

> What does exist in mathematics, is the operation restriction:

>

> f | A = { (x,y) e f | x e A }

>

> And you can for example say:

>

> id_X = id_Y | X

>

> for X ⊆ Y. And subsequetly you can also say id_Y | X is a X -> X bijection.

>

Thanks for clearing that up, Jan Burse. (Hee, hee!)

Sep 22, 2022, 5:32:17 PM9/22/22

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Nope, id_y | x is a bijection on x.

id_y isn't a bijection on x.

Whats wrong with you?

Dan Christensen schrieb:

id_y isn't a bijection on x.

Whats wrong with you?

Dan Christensen schrieb:

Sep 22, 2022, 6:56:36 PM9/22/22

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On Thursday, September 22, 2022 at 5:32:17 PM UTC-4, Mostowski Collapse wrote:

> Nope, id_y | x is a bijection on x.

> id_y isn't a bijection on x.

>

It is. Deal with it, Jan Burse.
> Nope, id_y | x is a bijection on x.

> id_y isn't a bijection on x.

>

Sep 22, 2022, 8:12:35 PM9/22/22

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Not in math text books. Did your mama

notice that you are brain damaged?

The word "is a" translates to membership,

so the two sentences translate to:

id_y | x e { f : x -> x | f bijective }

id_y e { f : x -> x | f bijective }

For x ⊆ y & x =\= y only the first membership

is true, the second membership is false.

Dan Christensen schrieb:

Sep 22, 2022, 10:30:51 PM9/22/22

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On Thursday, September 22, 2022 at 8:12:35 PM UTC-4, Mostowski Collapse wrote:

> For x ⊆ y & x =\= y only the first membership

> is true, the second membership is false.

>

> Dan Christensen schrieb:

> > On Thursday, September 22, 2022 at 5:32:17 PM UTC-4, Mostowski Collapse wrote:

> >> Nope, id_y | x is a bijection on x.

> >> id_y isn't a bijection on x.

> >>

> >

> > It is. Deal with it, Jan Burse.

> >

> For x ⊆ y & x =\= y only the first membership

> is true, the second membership is false.

>

> Dan Christensen schrieb:

> > On Thursday, September 22, 2022 at 5:32:17 PM UTC-4, Mostowski Collapse wrote:

> >> Nope, id_y | x is a bijection on x.

> >> id_y isn't a bijection on x.

> >>

> >

> > It is. Deal with it, Jan Burse.

> >

> Not in math text books.

[snip childish abuse]
>

> The word "is a" translates to membership,

> so the two sentences translate to:

>

> id_y | x e { f : x -> x | f bijective }

> id_y e { f : x -> x | f bijective }

>

Here is a the usual definition/abbreviation for the bijectivity predicate:

ALL(f):ALL(a):ALL(b):[Bijective(f,a,b) <=>

[ALL(c):ALL(d):[c in a & d in a => [f(c)=f(d) => c=d]] & ALL(c):[c in b => EXIST(d):[d in a & f(d)=c]]]

It's use is optional. It is used only to improve readability. Instead of writing the abbreviation Bijective(f,a,b), you could write it out in full as:

ALL(c):ALL(d):[c in a & d in a => [f(c)=f(d) => c=d]] & ALL(c):[c in b => EXIST(d):[d in a & f(d)=c]]

From the definitions of x, y and id_y above, we have Bijective(id_y,x,x). Deal with it, Jan Burse. Just admit you were wrong.

Sep 22, 2022, 11:27:42 PM9/22/22

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On Friday, September 23, 2022 at 4:30:51 AM UTC+2, Dan Christensen wrote:

> Here is a

nonsensical

> definition/abbreviation for the bijectivity predicate:

>

> ALL(f):ALL(a):ALL(b):[Bijective(f,a,b) <=>

> [ALL(c):ALL(d):[c in a & d in a => [f(c)=f(d) => c=d]] & ALL(c):[c in b => EXIST(d):[d in a & f(d)=c]]]

Yeah, it's nonsense.

> Here is a

nonsensical

> definition/abbreviation for the bijectivity predicate:

>

> ALL(f):ALL(a):ALL(b):[Bijective(f,a,b) <=>

> [ALL(c):ALL(d):[c in a & d in a => [f(c)=f(d) => c=d]] & ALL(c):[c in b => EXIST(d):[d in a & f(d)=c]]]

Sep 22, 2022, 11:50:47 PM9/22/22

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On Thursday, September 22, 2022 at 11:27:42 PM UTC-4, Fritz Feldhase wrote:

> On Friday, September 23, 2022 at 4:30:51 AM UTC+2, Dan Christensen wrote:

>

> > Here is definition/abbreviation for the bijectivity predicate:
> On Friday, September 23, 2022 at 4:30:51 AM UTC+2, Dan Christensen wrote:

>

> >

> > ALL(f):ALL(a):ALL(b):[Bijective(f,a,b) <=>

> > [ALL(c):ALL(d):[c in a & d in a => [f(c)=f(d) => c=d]] & ALL(c):[c in b => EXIST(d):[d in a & f(d)=c]]]

> Yeah, it's nonsense.

Oh, really? How do YOU define Bijective(f,a,b), Fritz?
> > ALL(f):ALL(a):ALL(b):[Bijective(f,a,b) <=>

> > [ALL(c):ALL(d):[c in a & d in a => [f(c)=f(d) => c=d]] & ALL(c):[c in b => EXIST(d):[d in a & f(d)=c]]]

> Yeah, it's nonsense.

Oct 6, 2022, 9:49:48 PM10/6/22

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And, for any b in B, there exists unique a in A such that F^-1(b) = a.

There exists F.

Here the unique part of course means for-any a, exists b, not exists

c =/= b, s.t. F(a) = c, and vice versa.

These days often set theorists are lazy and let Cantor/Schroeder/Bernstein thm.

work out there "exists" such an F, while of course it must result the above.

Oct 6, 2022, 9:52:44 PM10/6/22

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and it's pretty clear the rules are absolutely arbitrary.

Oct 6, 2022, 10:56:08 PM10/6/22

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You did say "function", right?

You got DesCartes, and for DesCartes, it was the time of geometry defining function.

So, the Cartesian function, it's the space of all the single value mappings from domain

to codomain. (Domain, image, domain, range, domain, codomain, image.)

This is where in geometry the only "space" is supports here is a line in geometry. According

to, "Euclid", the line is what exists and there are points on it. So, that's "not" a Cartesian function.

(I.e. functions like it without otherwise the multivalent are "discontinuous" functions.)

Which among them are series, ....

Oct 6, 2022, 11:25:02 PM10/6/22

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extra conditions will never reverse on me when taken the wrong way".

Also called "science".

Yeah, why don't you put "science" first in all your theorems.

The "science": part is only for the "non-logical",

the "logical" is already "logical".

Of course there's syllogism, where words are abstract non-logically.

Mostly "fulfills question words with respect to is or has".

Oct 6, 2022, 11:31:29 PM10/6/22

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Oct 6, 2022, 11:59:38 PM10/6/22

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>

> Yeah, why don't you put "science" first in all your theorems.

>

Can you give even a single example of an actual error in science, engineering or any other human endeavour arising strictly from the application of faulty rules of logic that you imagine? What were the consequences? How many lives were lost or ruined, etc.

Oct 7, 2022, 12:05:40 AM10/7/22

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It's about the shortest way to say it, ....

(... insert Wiki ....)

Oct 7, 2022, 12:18:15 AM10/7/22

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On Friday, October 7, 2022 at 12:05:40 AM UTC-4, Ross A. Finlayson wrote:

> >

> > Can you give even a single example of an actual error in science, engineering or any other human endeavour arising strictly from the application of faulty rules of logic that you imagine? What were the consequences? How many lives were lost or ruined, etc.

> >

> > Can you give even a single example of an actual error in science, engineering or any other human endeavour arising strictly from the application of faulty rules of logic that you imagine? What were the consequences? How many lives were lost or ruined, etc.

> If you ever forget that I'd feel sorry for you.

>

> It's about the shortest way to say it, ....

>

>

>

> (... insert Wiki ....)

So, not a single example. A tempest in a teacup, then. Whew!
>

> It's about the shortest way to say it, ....

>

>

>

> (... insert Wiki ....)

Dan

Oct 7, 2022, 12:19:23 AM10/7/22

to