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#1 2007-04-25 16:03:35
- Stanley_Marsh
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- Registered: 2006-12-13
- Posts: 345
Need proof.
Last edited by Stanley_Marsh (2007-04-25 16:05:42)
Numbers are the essence of the Universe
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#2 2007-04-25 17:28:00
- Ricky
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- Registered: 2005-12-04
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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
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- Registered: 2007-02-23
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Re: Need proof.
The standard way to prove that X is a subset of Y is to let x ∈ X and show that x ∈ Y. That is all.
Its as straightforward as that.
Heres an exercise for you: 
Last edited by JaneFairfax (2007-04-25 21:56:57)
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#4 2007-04-26 09:27:03
- Stanley_Marsh
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- Registered: 2006-12-13
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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
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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
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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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Re: Need proof.
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
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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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