#Need help please :)

161 messages · Page 1 of 1 (latest)

frank cradle
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Can someone please help me prove the last one

IMG_C8CB3169-7554-4A09-940C-8348014CB8F4.jpg
livid valeBOT
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molten stratus
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,rotate

opal ivyBOT
molten stratus
frank cradle
frank cradle
molten stratus
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dont spam ping, but no, it's just the set of clusters

frank cradle
molten stratus
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replies ping.

frank cradle
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Oh

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I’m sorry then

fallow crane
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$$ x\in D(X) \Leftrightarrow x\in \mathrm{Cl}(X\setminus {x}) $$

opal ivyBOT
molten stratus
molten stratus
frank cradle
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But where do I start

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By taking a point from D(A)?

molten stratus
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the contrapositive is $A'\cup B'\not\subseteq (A\cup B)'$

opal ivyBOT
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Omegabet_

186109587366215691.png
fallow crane
molten stratus
opal ivyBOT
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Omegabet_

186109587366215691.png
frank cradle
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X is in A U B

molten stratus
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' is derived set

frank cradle
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oh

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I’m so confused

molten stratus
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on?

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cause 'im so confused' doesn't help either of us

frank cradle
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if x doesn’t belong to D(A) or D(B) then it’s not a limit point

molten stratus
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or, not and

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but if x isnt in D(A)UD(B), then it's not in D(A) or in D(B)

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hence there exists neighborhoods of x such that...?

fallow crane
# frank cradle I’m so confused

when you prove inclusion A subset B, you have two options:

  1. take x in A and show x in B
  2. take x notin B and show x notin A

you need to appeal to the notions of "derived set" and "set union" for this problem

frank cradle
fallow crane
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are you asking or telling?

frank cradle
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both

molten stratus
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Yes, there exists a neighbourhood of $x$, $U_x$, such that $(U_x\cap A)\setminus{x}=\emptyset$

opal ivyBOT
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Omegabet_

186109587366215691.png
frank cradle
molten stratus
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Likewise V_x for B

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now consider $G:=U_x\cap V_x$. This is an open neighbourhood of $x$, and one (being you) can show that $(G\cap(A\cup B))\setminus{x}=\emptyset$

opal ivyBOT
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Omegabet_

186109587366215691.png
frank cradle
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U_x and V_x are two neighborhoods?

molten stratus
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yes

frank cradle
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and G is the open set

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that contain both

molten stratus
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G is the intersection of the neighbourhoods provided to us by $x\not\in D(A)$ and $x\not\in D(B)$

opal ivyBOT
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Omegabet_

186109587366215691.png
molten stratus
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ie unpacking the definition of not being a cluster point gives us these U_x and V_x for A and B respectively

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I claim that G only intersects with AUB at x, which is now just an exercise in some set theory

fallow crane
# opal ivy **aL**

alternatively, apply this and the fact that closure of finite union is the finite union of closures

molten stratus
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anyway I have to go study for a quiz, so maybe now aL can take over without being in the way

fallow crane
frank cradle
molten stratus
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ty

frank cradle
fallow crane
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sound good

frank cradle
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and I haven’t revised closure yet so I can’t use it here until I get to that chapter

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This is what we just talked about

frank cradle
frank cradle
fallow crane
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this is confusing me

image.png
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what do we assume?

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are you proving the contrapositive?

frank cradle
fallow crane
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a limit point of what?

frank cradle
frank cradle
fallow crane
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alright, and you want to prove x is not a limit point of A union B

frank cradle
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yes i guess

fallow crane
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what does it mean x is not a limit point of X by the definition you are working with?

frank cradle
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That x doesn’t belong to derived set A or B

fallow crane
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how do you define derived set

frank cradle
fallow crane
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, rotate

opal ivyBOT
fallow crane
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ok so, put all of that together

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"x is not a limit point of A"

there exists a neighborhood of x, call it V_x, such that

opal ivyBOT
frank cradle
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it can also be phi

fallow crane
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in that case

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$$ V_x\cap (A\setminus {x}) = \emptyset $$

frank cradle
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true

opal ivyBOT
fallow crane
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so, now assume x is notin D(A) and x is notin D(B)

frank cradle
frank cradle
fallow crane
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you have two neighborhoods of x now

frank cradle
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so

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Since they are not in the union

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and not in them

fallow crane
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write out what it means x is not a limit point of A union B

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and figure out whether you have what you need for it

frank cradle
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can we use complements

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So we can change the union to intersection

fallow crane
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play around with it, assume you have the following

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$$ A_x \cap (A\setminus {x}) = \emptyset = B_x \cap (B\setminus {x}) $$

opal ivyBOT
frank cradle
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I will go and think and tell you what I got

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hopefully I will get it this time_catnerd

fallow crane
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to show x is not a limit point of A U B, it suffices to find a neighborhood of x , C_x such that

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$$ C_x \cap ((A\cup B)\setminus {x}) = \emptyset$$

opal ivyBOT
frank cradle
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That c is closure ?

fallow crane
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no

frank cradle
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it’s another neighborhood

fallow crane
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yes

frank cradle
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We have three now

fallow crane
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no, we have two

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your task is to figure out how to obtain C_x from the assumptions you have

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I only wrote out what it means by definition that x is not a limit point of A U B

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i didn't claim it was actually true

frank cradle
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okay so I’m thinking now

fallow crane
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take your time

frank cradle
frank cradle
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This what I got

IMG_5727.jpg
frank cradle
fallow crane
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don't write that, it's not always correct, that was my bad

image.png
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stick to this version

image.png
frank cradle
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Al but that is too complicated for me

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If I don’t use the contrastive can I do it the normal way like this way

fallow crane
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you can

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assume x is a limit point of A U B, x is not a limit point of A and prove x is a limit point of B

frank cradle
fallow crane
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, rotate

frank cradle
opal ivyBOT
fallow crane
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that is correct

frank cradle
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THANK YOU SO MUCH AL

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YOU HELPED ME SO KYCH

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much*

fallow crane
frank cradle
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Just write down what was written

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in the question

fallow crane
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what do you think?

frank cradle
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I think I should write therefore D(A U B)=D(A)UD(B)

fallow crane
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the converse inclusion is obvious, so that works

frank cradle
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so DONE?

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again tysmmm Al you are so smart

fallow crane
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you did all the work

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im just chilling playing new vegas kekbusiness

frank cradle
frank cradle
fallow crane
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something for a nerd like me kekbusiness

frank cradle
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OOO

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NERD

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That is so attractive

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+close

simple cedarBOT
# frank cradle +close
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