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 April 3rd, 2012, 09:28 PM #1 Newbie   Joined: Apr 2012 Posts: 2 Thanks: 0 Logic Hi, I have this problem whereby I need to convert a sentence into a logic statement. It'd be great if I could get some feedback on where I am going wrong/right. The sentence to translate: Let $x,y,\epsilon \in \mathbb{R},$ if $(x \le y + \epsilon)$ for every $\epsilon > 0$ then $(x \le y )$. I have come up with two possible translations: 1. Let $x,y,\epsilon \in \mathbb{R}, \ \forall \epsilon [(\epsilon>0) \rightarrow (x \le y + \epsilon)] \rightarrow (x \le y)$ 2. Let $x,y,\epsilon \in \mathbb{R}, \ \forall \epsilon (\epsilon>0) \rightarrow [(x \le y + \epsilon) \rightarrow (x \le y)]$ I am leaning towards my first translation. What does everyone else think? Thanks.
 April 3rd, 2012, 09:38 PM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Logic Yes, the first one is right. The second one says something else which happens to be correct: if x <= y + e for e > 0, then x <= y.
 April 3rd, 2012, 09:46 PM #3 Newbie   Joined: Apr 2012 Posts: 2 Thanks: 0 Re: Logic Thank you so much CRGreathouse!
April 4th, 2012, 03:04 AM   #4
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Re: Logic

Quote:
 Originally Posted by CRGreathouse Yes, the first one is right. The second one says something else which happens to be correct: if x <= y + e for e > 0, then x <= y.

Only a formal proof justifies the correctness of a formula ,since there are not rules in tranlating from words to symbols,you should know that.

And formulas in mathematics are closed formulas,so technicaly speeking none of the formulaes of the OP are correct

April 4th, 2012, 06:03 AM   #5
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Re: Logic

Quote:
 Originally Posted by orokusaki Hi, I have this problem whereby I need to convert a sentence into a logic statement. It'd be great if I could get some feedback on where I am going wrong/right. The sentence to translate: Let $x,y,\epsilon \in \mathbb{R},$ if $(x \le y + \epsilon)$ for every $\epsilon > 0$ then $(x \le y )$. I have come up with two possible translations: 1. Let $x,y,\epsilon \in \mathbb{R}, \ \forall \epsilon [(\epsilon>0) \rightarrow (x \le y + \epsilon)] \rightarrow (x \le y)$ 2. Let $x,y,\epsilon \in \mathbb{R}, \ \forall \epsilon (\epsilon>0) \rightarrow [(x \le y + \epsilon) \rightarrow (x \le y)]$ I am leaning towards my first translation. What does everyone else think? Thanks.

Why should the 1st one be the correct one?

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