Cont: Deeper than primes - Continuation 2

[doronshadmi]

No. There is an injection for which A (not its cardinality) is the domain and B (not its cardinality) is the codomain.

Where exactly do you see that I do not distinguish between A or B (sets) and |A| or |B| (their cardinalities)?

There is no basis for your "No."

Nope, not that last part.

Yep, yes that last part, since injection is not the same as bijection.

 

[doronshadmi]

Nothing in my definition for cardinality is affected by whether the sets are finite or infinite.

You don't have definition for cardinality, as shown in http://www.internationalskeptics.com/forums/showpost.php?p=13126710&postcount=3478.

(Some correction of (1) in http://www.internationalskeptics.com/forums/showpost.php?p=13126710&postcount=3478

It has to be:

1) The notation |A| <= |B| means that |A| is less than or equal to |B| (or, equivalently, at most |B|, or not greater than |B|)

 

[jsfisher]

Where exactly do you see that I do not distinguish between A or B (sets) and |A| or |B| (their cardinalities)?

There is no basis for your "No."

It was an error. I had read that part of your post too quickly and misinterpreted what you wrote.

Yep, yes that last part, since injection is not the same as bijection.

You might want to recheck the meaning of "injection". One-to-one is a requirement. Onto doesn't matter.

Not all injections are bijections, but all bijections are injections.

 

[doronshadmi]

Not all injections are bijections, but all bijections are injections.

No injection is bijection, unless injection (one-to-one) and surjection (onto) are combined.

One-to-one AND onto (bijections) can't be one-to-one (injections).

 

[jsfisher]

One-to-one AND onto (bijections) can't be one-to-one (injections).

Why not? Are odd numbers forbidden to be odd when they happen to be prime?


A mapping that is one-to-one is an injection.
A mapping that is onto is a surjection.
A mapping that is both an injection and a surjection is a bijection (and still an injection and still a surjection).

 

[doronshadmi]

Why not? Are odd numbers forbidden to be odd when they happen to be prime?

Being an odd number does not exclude it of from the set of prime numbers.

A mapping that is one-to-one is an injection.
A mapping that is onto is a surjection.
A mapping that is both an injection and a surjection is a bijection (and still an injection and still a surjection).

Thank you for clarify my argument that invective mapping is not the same as bijective mapping, exactly because bijection is not less than injection AND surjection.



So we are back to "Cardinality of a set is a measure of the number of members of the set" and no two sets must be used in order to know that.

Examples in case of finite (strict) cardinalities:

|{}|=0
|{{}}|=1
|{{},{{}}}|=2

An example of a non-finite (non-strict) cardinality:

|{
|{}|=0,
|{{}}|=1,
|{{},{{}}}|=2,
...
}|

where ... is a notation for a set with non-strict cardinality.

 

[jsfisher]

So we are back to "Cardinality of a set is a measure of the number of members of the set" and no two sets must be used in order to know that.

Does this mean you have abandoned without retraction your blunder about bijections not being injections? No matter. So, I see you are abandoning my definition for cardinality in favor of your own. Fine, I suppose.

In my version, I told you explicitly how measurement worked: It was by comparison of two sets using injective mappings as the basis of the comparison. |A| <= |B| if and only if there is an injection from A to B.

In your version, how do you propose to "measure the number of members of the set"? (A simplistic response of "just count them" earns you no points. You need to be clear how your measure works.)

 

[Little 10 Toes]

Once again, you start using non-standard notation.

Cardinality is the size of a set. It's not that hard.

 

[jsfisher]

Doronshadmi,
I see you continue your dishonest practice of grossly changing the content of your posts after they have been responded to. Nevertheless...

Being an odd number does not exclude it of from the set of prime numbers.

And something being red AND round doesn't stop it from being red.
And a mapping being bijective doesn't stop if from being injective.

Thank you for clarify my argument that invective mapping is not the same as bijective mapping, exactly because bijection is not less than injection AND surjection.

That was not your argument. Your argument was

One-to-one AND onto (bijections) can't be one-to-one (injections).

and that statement is patently false. Have the decency to own it and admit the error.

 

[doronshadmi]

And something being red AND round doesn't stop it from being red.

Yet being red is not the same as being red AND round.

 

[doronshadmi]

and that statement is patently false. Have the decency to own it and admit the error.

You are right my mistake, it has to be: One-to-one AND onto (bijections) can't be only one-to-one (injections).

 

[doronshadmi]

Does this mean you have abandoned without retraction your blunder about bijections not being injections? No matter. So, I see you are abandoning my definition for cardinality in favor of your own. Fine, I suppose.

In my version, I told you explicitly how measurement worked: It was by comparison of two sets using injective mappings as the basis of the comparison. |A| <= |B| if and only if there is an injection from A to B.

In your version you have told me that injection or bijection have no difference in mapping from A to B. Moreover You have claimed that strict equality and strict inequality follow directly from your definition, but <= is a non-strict inequality.

In your version, how do you propose to "measure the number of members of the set"? (A simplistic response of "just count them" earns you no points. You need to be clear how your measure works.)

Cardinality is a measure of the number of members of set A, such that (A is finite iff |A| is strict) OR (A is non-finite iff |A| is non-strict).

 

[Little 10 Toes]

Define "strict" and "non-strict".

 

[Little 10 Toes]

Dupe.

 

[jsfisher]

In your version you have told me that injection or bijection have no difference in mapping from A to B.

I never said that.

Moreover You have claimed that strict equality and strict inequality follow directly from your definition, but <= is a non-strict inequality.

...and I told you how they do.

Cardinality is a measure of the number of members of set A, such that (A is finite iff |A| is strict) OR (A is non-finite iff |A| is non-strict).

(1) You now need to define "strict" and "non-strict" as they apply to sets. So far your usage has been indistinguishable from strict meaning finite and non-strict being non-finite. The renders your "such that" meaningless.

(2) How does one conclude anything from your claimed definition of cardinality? You have provided no measure whatsoever, be it a relative or absolute measure. Moreover, how can you tell if a set is finite or not?

 

[jsfisher]

You are right my mistake, it has to be: One-to-one AND onto (bijections) can't be only one-to-one (injections).

So, you are proposing to change your prior response from one that was completely wrong to one that is completely irrelevant?

Here was the context which you now attempt to conveniently spin out of existence:

Not all injections are bijections, but all bijections are injections.

...
One-to-one AND onto (bijections) can't be one-to-one (injections).

 

[jsfisher]

Terms Doronshadmi uses in undefined ways:

• accurate value
• accurate amount
• strict value
• cardinality
• accurate cardinality
• infinite set -- superseded by finite and non-finite, below
• complete whole
• complete set
• next object
• Biggest result
• Strict and non-strict
• Finite and non-finite

By the way, that last pair, finite and non-finite, follows directly from cardinality, but you need a definition for cardinality, first.

 

[jsfisher]

Once again, you start using non-standard notation.

Cardinality is the size of a set. It's not that hard.

It is that hard if you are trying to back into it from the result you want.

 

[doronshadmi]

I never said that.

yes you did

You might want to recheck the meaning of "injection". One-to-one is a requirement. Onto doesn't matter.

Not all injections are bijections, but all bijections are injections.

Since injection AND surjection (bijection) is not the same as injection-only, not even a single bijection is injection-only AND not even a single injection-only is bijection, so "Not all injections are bijections, but all bijections are injections." is false.

...and I told you how they do.

Wrong, since <= is a non-strict inequality, no "meanings for strict equality and strict inequality of cardinalities follow directly" from your, so called, definition.

(1) You now need to define "strict" and "non-strict" as they apply to sets.

Ok.

If for bijection f : A → { 0 , … , n } |A| is some n, then |A| is called strict.

Cardinality is a measure of the number of members of set A, such that (A is finite iff |A| is strict) OR (A is non-finite iff |A| is non-strict).

By non-strict I mean that |A| is not any some n, since the number of members of set A is changed by successor operation.

You have provided no measure whatsoever, be it a relative or absolute measure.

I have provided two types of measures of the number of members of set A, where one provides a strict measure of the number of members of set A , and the other provides a non-strict measure of the number of members of set A.

 

[jsfisher]

yes you did

Since injection AND surjection (bijection) is not the same as injection-only, not even a single bijection is injection-only AND not even a single injection-only is bijection, so "Not all injections are bijections, but all bijections are injections." is false.

It is much easier to find mistakes in the statements of others if you free change what was actually said. I made no mention of "injection-only" -- that was your bit of foolishness.

Not all closed curves are circles, but all circles are closed curves. Not all injections are bijections, but all bijections are injections.

My statement remains correct. Your accusation I said something I did in fact not say remains incorrect.

Wrong, since <= is a non-strict inequality, no "meanings for strict equality and strict inequality of cardinalities follow directly" from your, so called, definition.

Why not? This is standard stuff, Doronshadmi. From any standard sort of meaning of the <= relationship, one can readily derive meanings for < and =.
You did not learn this?

Ok.

If for bijection f : A → { 0 , … , n } |A| is some n, then |A| is called strict.

Well, that's a very limited definition for cardinality. Got a definition that works for all sets? And it is now clear you are just using "strict" to make it sound official or something.

It is also clear you are building your conclusion into your definitions.

As always, you are free to develop your whole branch of doronetics, with whatever special definitions you like, but you don't get to drag your nonsense into a discussion of normal Mathematics just to claim it is wrong.

It isn't, but you are.