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#1 2014-05-01 20:49:53

Agnishom
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Set Theory Identity

Prove that:


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
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#2 2014-05-01 21:00:54

Nehushtan
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Re: Set Theory Identity

What does
mean?


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#3 2014-05-01 21:24:51

Agnishom
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Re: Set Theory Identity

Fo7a2Z9.jpg


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#4 2014-05-01 21:56:05

bob bundy
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Re: Set Theory Identity

Looks to me like you have 'proved' it with the diagrams.  All you need to do is convert that into set statements and it is done.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#5 2014-05-01 22:05:17

Agnishom
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Re: Set Theory Identity

What are set statements?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#6 2014-05-01 22:31:41

Nehushtan
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From: London
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Posts: 615
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Re: Set Theory Identity

Something is wrong with your question.


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#7 2014-05-01 22:43:43

Agnishom
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Re: Set Theory Identity

Sorry, it is supposed to be This:
Prove that:

I can figure it out myself if you can explain how to get the third line fromt the second

Last edited by Agnishom (2014-05-01 22:47:05)


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#8 2014-05-01 22:51:15

bob bundy
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Registered: 2010-06-20
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Re: Set Theory Identity

Hmmm.  I'm getting Nehushtan's result too, using your definition:

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#9 2014-05-01 22:52:02

bob bundy
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Re: Set Theory Identity

Ah! just seen your correction.  Revised version coming up.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#10 2014-05-01 22:52:50

Nehushtan
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Re: Set Theory Identity

Agnishom wrote:

Sorry, it is supposed to be This:
Prove that:

That’s better. smile

and it is easy to see that

.

Last edited by Nehushtan (2014-05-01 22:56:57)


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#11 2014-05-01 22:56:03

bob bundy
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Re: Set Theory Identity

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#12 2014-05-01 22:57:12

Agnishom
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Re: Set Theory Identity

Nehushtan,

I see it all. The problem seems to be with the proof.

Please Tell me how I can show

bob,

Sorry, I do not get it.

Last edited by Agnishom (2014-05-01 22:58:14)


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#13 2014-05-01 23:04:00

bob bundy
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Re: Set Theory Identity

Oh. 

line 1: I used your definition for the triangly thing.

line 2: I simplified it.  Couldn't make a dash work as in A-dash = not A so I used a wobbly over tilde thingy.

phi is the empty set.

As for this:

Use the distributivity laws

In A AND in the union = just A

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#14 2014-05-01 23:12:42

Agnishom
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Re: Set Theory Identity

What did you simplify in line 2? How is this relevant?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#15 2014-05-02 00:23:45

bob bundy
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Registered: 2010-06-20
Posts: 6,468

Re: Set Theory Identity

I'll try it in more detail:

So A big triangled with the intersection is (A less the intersection) united with (the intersection less A)

A less  the intersection is the bits of A without the bits in B.

The intersection without A is nothing at all so the empty set.

Unite these.

But everything in a set plus the empty set is still just that set so

This is just the definition of \, hence

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#16 2014-05-02 00:28:45

Nehushtan
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From: London
Registered: 2013-03-09
Posts: 615
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Re: Set Theory Identity

Agnishom wrote:

Please Tell me how I can show


Note that
is a subset of
. In general, if
then
.

Proof:

We must prove that both

and
.

is clear since
is a subset of
.

To prove the other inclusion, suppose

and let
.
. If
we are done. If
then, since
, we have
as well. In either case
. Hence
.


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#17 2014-05-02 02:11:09

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 18,005
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Re: Set Theory Identity

Hmm..

Thanks,


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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