Proof using WOP

[kmerfeld]

Hello everyone,

 

I want to prove that for postive integers x, y, y is not equal to x+y. I want to do this using the WOP. Here's what I have done so far:

 

Suppose for some postive integer x, there exists a y st y y+x. By the WOP, there exists a smallest x₀ st y=x₀+y. Now I think I may have to apply the WOP again, but am not sure. Any advice?

 

Thanks a lot,

Kevin

 

P.S. A little help using special BBC code. I'd like to use LaTex but when I try to do the not equal sign it shows up as:

 

[latex]\ne
[/latex]

 

How do I get rid of the < br > ?

Edited March 11, 2013 by kmerfeld

[Klaynos]

 

[latex]\ne[/latex]

 

How do I get rid of the < br > ?

Remove the return before the closing tag.

[kmerfeld]

[latex]\ne[/latex]

 

There she goes. Thanks

Edited March 11, 2013 by kmerfeld

[Nehushtan]

Hello everyone,

 

I want to prove that for postive integers x, y, y is not equal to x+y. I want to do this using the WOP. Here's what I have done so far:

 

Suppose for some postive integer x, there exists a y st y y+x. By the WOP, there exists a smallest x₀ st y=x₀+y. Now I think I may have to apply the WOP again, but am not sure. Any advice?

Apply the WOP to [latex]y[/latex], not [latex]x[/latex]. First, however, show that there does not eixst any positive integer [latex]x[/latex] such that [latex]1=1+x[/latex]. For this you can make use of the axiom that 1 is not the successor element of any positive integer.

[kmerfeld]

Hello Nehushtan,

 

Thanks for the suggestion. However, the book requires that I first apply the WOP on x. Then it "hints" that I should apply the WOP again...

 

KM

[kmerfeld]

I think I figured it out:

 

I want to show [latex]y \ne x+y [/latex] for all positive integers x and y.

 

So suppose there exists a y such that [latex]y=x+y[/latex]

 

Then by WOP, there exists a smallest x such that:

 

[latex]y=x_{0} +y[/latex]

 

Again by WOP, there exists a smallest y such that:

 

[latex]y_{0}=x_{0}+y_{0}[/latex]

 

Then:

 

[latex]y_{0}-y_{0}=x_{0}[/latex]

 

 

[latex]0=x_{0}[/latex]

 

But 0 is not a positive number. So this is a contradiction, meaning:

 

 

[latex]y \ne x+y [/latex] for all positive integers x and y.

 

Now it does seem like I could have done this proof without using WOP, but it still seems to work this way, and this is how I was asked to do it.

 

Thanks everyone. Consider question this solved.