From owner-cse191-sp08-list@LISTSERV.BUFFALO.EDU  Wed Apr 30 11:41:41 2008
Date:         Wed, 30 Apr 2008 11:41:15 -0400
From: "William J. Rapaport" <rapaport@cse.Buffalo.EDU>
Subject: 191 Question about FOPL
To: CSE191-SP08-LIST@LISTSERV.BUFFALO.EDU

A student asks:
>     Would you please explain why these two propositions are different?  
> They are parts b and c of  #6  on the mid-term.  Thank you.
>    
>     EnAm[m<=n]	(1)
> 
>     and
> 
>     AmEn[m<=n]	(2)

First, "m<=n" says that m is smaller than (or equal to) n,
i.e., that n is bigger than (or equal to) m.


(1) says:  "there exists a number n such that, for any number m,
		m <= n"

i.e.:	there is a number n that has the following property:
	for any number m, m <= n

i.e.:	there is a number n that is bigger than every number.

That's false, of course.


(2) says:  "for any number m, there is a number n such that m <= n"

i.e.:	Choose any number m that you want.
	Then you can find a number n that is bigger than it.	

That's true:  Just take n = m+1.  In fact, because this uses "<=", you
could just let n=m.

That's why they're different:  They have different truth values.