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 Post subject: Smallest Element proofPosted: Thu, 12 Jan 2012 04:26:43 UTC
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Joined: Mon, 1 Nov 2010 17:13:40 UTC
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Hey all, need some help with the following proof:

Let A be a non-empty subset of Z and b ∈ Z, such that for each a ∈ A, b <= a. Then A has a smallest element.

All help appreciated!

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 Post subject: Re: Smallest Element proofPosted: Thu, 12 Jan 2012 04:39:46 UTC
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jstarks4444 wrote:
Hey all, need some help with the following proof:

Let A be a non-empty subset of Z and b ∈ Z, such that for each a ∈ A, b <= a. Then A has a smallest element.

All help appreciated!

Use the corresponding result of .

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 Post subject: Re: Smallest Element proofPosted: Thu, 12 Jan 2012 05:37:11 UTC
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An added hint to go with outermeasure's suggestion:

Consider the set which is a subset of , and you know .

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 Post subject: Re: Smallest Element proofPosted: Sun, 15 Jan 2012 17:51:48 UTC
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Joined: Mon, 1 Nov 2010 17:13:40 UTC
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is there a way to use the Well-Ordering Principle and "construction" here?

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 Post subject: Re: Smallest Element proofPosted: Sun, 15 Jan 2012 19:57:56 UTC
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jstarks4444 wrote:
is there a way to use the Well-Ordering Principle and "construction" here?

What do you mean? Both proofs we talked about make use of the well-ordering on ...And what exactly is "construction"

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 Post subject: Re: Smallest Element proofPosted: Sun, 15 Jan 2012 23:49:40 UTC
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 Post subject: Re: Smallest Element proofPosted: Mon, 16 Jan 2012 05:36:17 UTC
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Joined: Mon, 1 Nov 2010 17:13:40 UTC
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Why can't 'b' be negative?

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 Post subject: Re: Smallest Element proofPosted: Mon, 16 Jan 2012 07:31:41 UTC
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jstarks4444 wrote:
Why can't 'b' be negative?

b can be negative. Where did we state it isn't?

That's why you reduce it to by translating by -b+1 (or -b, depending on your definition of ), and not just (incorrectly!) assume everything you have is nonnegative.

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