Injectivity

[tepid cairn]

How do I write the function for 4 f

[ruby basin]

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[tepid cairn]

@valid crag

[valid crag]

Aight so what are the elements in $(A^{\omega})^n$

[stoic walrus]

𝐍𝐢𝐠𝐞𝐥𝐚𝐭𝐞𝐝 𝐕𝐞𝐠𝐞𝐭𝐚

[valid crag]

What do they look like

[tepid cairn]

a1^n,a2^n......,

Something like this?

[valid crag]

Huh

[tepid cairn]

Wait

[valid crag]

It's a n tuple of sequences in A

Right?

[tepid cairn]

A^w represents countably infinite elements

[tepid cairn]

Yes

[valid crag]

Alright

So now let's first do this for n=2

Okay?

If you have two sequences (a_k) and (a'_k) in A

Can you interweave them into a new sequence

[tepid cairn]

like a1,a'1, a2,a'2,....?

[valid crag]

Yeahh

So now this gives you a map to B^{omega} yes?

[tepid cairn]

Yes it does

[valid crag]

Is it injective?

[tepid cairn]

Wait

I understand for n = 2

But how would we do it till nth term

[valid crag]

Same idea

[tepid cairn]

We'll make n sequences?

[valid crag]

a_11a_12a_13..a_1n, a_21,a_22,...a_2n

[tepid cairn]

Like a^1 k, a^2 k,.....,a^2 k?

[tepid cairn]

Alright

[valid crag]

So now why is it injective

[tepid cairn]

We can use the property assuming f(a) = f(b) and showing a = b

[valid crag]

Can you reconstruct the n tuple from an element in the image?

[tepid cairn]

How do I reconstruct it

I don't know

How is reconstructing helpful

[valid crag]

We are buliding an inverse map from the img to the n tuple

So it's a bijection onto it's image

Anyways

The first n terms of f(a) would give you the first term of each sequence in the n tuple yeah?

[valid crag]

.

[tepid cairn]

Is the inverse mapping correct?

[valid crag]

The inverse mapping is not from B but from Img(f)

[tepid cairn]

I don't understand how it'd look

[valid crag]

So given $(b_i) \in img(f)$
$(b_i) \mapsto ((b_{nk-1}),(b_{nk-2}),...(b_{nk}))$

[stoic walrus]

𝐍𝐢𝐠𝐞𝐥𝐚𝐭𝐞𝐝 𝐕𝐞𝐠𝐞𝐭𝐚

[valid crag]

Basically write an i in n as i=nk+r where 0<=r < n

Then depending on the value of r send it to the rth place in the tuple

Makes sense?

[tepid cairn]

Somewhat

Makes sense ig

Digesting it

[valid crag]

Let's do this for n=2

For n=2 you'd separate

b_i into even and odd places

Then make a sequence out of all the even places and a sequence out of all the odd places

Then make two tuples

Similarly for n=k you look at the sequence mod k , I mean the index of the sequence mod k

[tepid cairn]

Is it possible for you to please show me this on paper I can't grasp it

[valid crag]

And make the tuples accordingly

[valid crag]

Okay gimme a minute

[tepid cairn]

Thanks

[valid crag]

\LARGE The last map is
$(a_1,a_2,...,a_n,a_{n+1},a_{n+2},...,a_{n+n},...) \mapsto \left[(a_1,a_{n+1},a_{2n+1},...),(a_2,a_{n+2},a_{2n+2},...), \cdots (a_n,a_{n+n},a_{2n+n},....)\right]$

[stoic walrus]

𝐍𝐢𝐠𝐞𝐥𝐚𝐭𝐞𝐝 𝐕𝐞𝐠𝐞𝐭𝐚

[tepid cairn]

Alright

I guess I'll have to reread this Convo multiple times to understand it

Ig that's my limitations

[valid crag]

Okay fine let's just do definitions then

[valid crag]

Let's prove injectivity by what you were suggesting

[valid crag]

.

Now given sequences f(a)=f(b)

Where a,b are tuples of sequences in A

Then they are equal in each index of f(a),f(b)

Right?

[tepid cairn]

Yes

[valid crag]

So this means they are equal in the first n indexes as well right?

[tepid cairn]

Yes

[valid crag]

Now what do the first in indexes mean hede

Here

Remember how our map was

[valid crag]

.

[tepid cairn]

Yes I remember

[valid crag]

Yeah so what are the first n indexes of a sequence in the image

How do they relate to a tuple in its preimage

[tepid cairn]

You tell me

[valid crag]

the first n indexes were each the first coordinates

Of the sequences in the tuple

Right?

[tepid cairn]

Yes

?

[valid crag]

You're doing it for n=2?

[tepid cairn]

Just wrote the first 2

To ask if I'm doing it correctly

[valid crag]

Uh idts

[tepid cairn]

Ohh

[valid crag]

Wait let me get my tablet and color code things maybe that would be easier for you to understand

[tepid cairn]

Okayy I'll check it in a while

[glass marsh]

It seems you are all set

[tepid cairn]

Thanks bro

I understand it now ig

@valid crag

[valid crag]

Fr?

[tepid cairn]

For for real

I'll solve it myself and send the soln just in case

[valid crag]

Nah it's okay

I'm glad you understand it now

[tepid cairn]

Thanks for all the efforts man

No prof in my uni would've went this far

[ruby basin]

@tepid cairn

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[tepid cairn]

+close

[ruby basin]

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