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#1 2007-04-25 16:03:35

Stanley_Marsh
Member
Registered: 2006-12-13
Posts: 345

Need proof.

 

Last edited by Stanley_Marsh (2007-04-25 16:05:42)


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#2 2007-04-25 17:28:00

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: Need proof.

1.  Here is an outline of the proof.

Take the area and divide it in half.  At least one of those halves has an infinite amount of point.  Now take this half area, and divide it in half.  Again, at least one quarter must contain an infinite amount of points.  Continue this infinitely.

Of course, an actual proof take a lot more work than that.

2.  Let x be in _ intersect _, and f(x) = y.  Then x is in _ and _, and so y is in _ intersect _.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#3 2007-04-25 21:53:16

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Need proof.

The standard way to prove that X is a subset of Y is to let xX and show that xY. That is all.

It’s as straightforward as that.

Here’s an exercise for you: smile

Last edited by JaneFairfax (2007-04-25 21:56:57)

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#4 2007-04-26 09:27:03

Stanley_Marsh
Member
Registered: 2006-12-13
Posts: 345

Re: Need proof.

Hmmm, 

Last edited by Stanley_Marsh (2007-04-26 09:28:46)


Numbers are the essence of the Universe

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#5 2007-04-26 11:24:51

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Need proof.

Be careful with that first line.

Let x is in A intersect B, and y be such that y = f(x).  Then y is in f(A intersect B).


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#6 2007-04-26 12:44:08

Stanley_Marsh
Member
Registered: 2006-12-13
Posts: 345

Re: Need proof.

Oh yah , wrong order ha!


Numbers are the essence of the Universe

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#7 2007-04-26 20:56:04

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Need proof.

Stanley_Marsh wrote:

You did not mention the injectivity of f. f must be injective, otherwise the reverse inclusion would not generally hold. For example:

Last edited by JaneFairfax (2007-04-27 03:02:27)

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#8 2007-04-27 02:10:57

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Need proof.

Specifically, it breaks down when you say that since y is in F(A) intersect F(B), then it must be that x is in A and B.  It may be the case that x is in A, z is in B where f(z) = y as well.  Then x need not be in B.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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