Conference rusure::math

I think not.

	Let's assume 1 = 0, and let's try to derive 1 = 1.

	1 = 0		Given
	0 = 1		Reflexive law
	1 = 1		Associative law (i.e., 1 = 0 = 1)

    See also note 450.
    

    re. .2
    
    It is clear to the common man on the street that some of the reasoning
    and the proofs given in 450 and other places do NOT comply with
    the rules of mathematics, but I think these proofs will make the
    reader to ask questions and may even teach the correct way by
    adentifing the wrong way.  
    
    All of us know that if we have  
    
         1 + 2  =  3
    
           3    =  3
    
    that the left side is equal the right side. So what? this is easy.
    But I am interested to see when the left side is not equal the right
    side of an equal sign (i.e  A = B, and A is not the same as B).
    
    
    
    

    re. .0
    
    Returning back to the problem suggested in .0 (i.e.
    
    can we prove a trigonometric identity by deriving  1 = 1, and in
    fact caqn we derive 1 = 1 if we assume 1 - cos ( x ) = sin ( x )
    )
    
    the answer is YES.
    
    here is the proof:
    
                  1 - cos ( x ) = sin ( x )
    
    multiply both sides by ( 1 + cos( x ) )
    
                1 - cos^2 ( x ) = sin( x ) * (1 + cos( x ))
    
    or   
                    sin^2 ( x ) = sin( x ) * (1 + cos( x ))
                                                    
    divide by  sin( x ) 
    
                      sin ( x ) = 1 + cos ( x )
      
    but we assumed that  sin( x ) = 1 - cos ( x )
    
    multiplying we get
    
                     sin^2( x ) = 1 - cos^2( x )
    
    that is 
    
                     sin^2( x ) = sin^2( x )
    
    devide by sin^2( x )
    
                              1 = 1
    
    What does this prove?
    
    It illustrates the fact that an incorrect assumption may be manipulated
    into a correct result. Just that, nothing more.
    
    Note:
         Do not use this reasoning in the real world. At the same time
    do not deposit infinite amount of money in the bank, because the
    interest you may get may not change your amount of money.
    
    Enjoy,
    
    Kostas G.
    <><><><><>               

    Re .4:
    
    Why go about things the long way?  Assume a = b.  If you like, a
    may be 1-cos(x) and b may be sin(x).  Since a = b, we can divide
    the left side of an equation by a and the right side by b.  So let's
    apply that to a = b.  a/a = b/b, so 1 = 1.
    
    There have been times when I have used or seen used a method which
    starts with an equation and works "backwards" to an identity.  The
    thing to remember here is that _this_is_not_a_proof_.  It is more
    of an exploration.  It becomes useful if one makes sure that each
    of the steps used is valid when performed in the opposite direction,
    because then it becomes possible, when the identity is finally found,
    to start with the identity and reverse all the steps to get to the
    original equation.  Then you have a proof.
    
    In the above, the irreversible step is assuming a = b.  After we
    reach 1 = 1 and try to reverse the steps, there is no way to justify
    assuming a = b.
    
    
    				-- edp

    re. .5
    
    I like your method and your reasoning.
    
    Well done.
    
    
    Kostas G.
    

                <<< TOOLS::ULTRIX$:[NOTES$LIBRARY]MATH.NOTE;1 >>>
                           -<  Mathematics at DEC  >-
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Note 467.7       Proving a trigonometric identity by "1 = 1" ?            7 of 7
EAGLEA::BEST "R. D. Best, 32 bit sys. arch. & A.D.," 49 lines   9-JUL-1986 22:56
                         -< invalid inference pattern >-
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>         Can we prove a trigonometric identity by deriving  1 = 1  from
>   it? Or more generally, can we prove something by assuming it and
>   deriving an incontestable result from it, using perfectly logical
>   steps all the way through?
    
>   For example, can we prove  
    
>      1 - cos( x ) = sin( x )  

>   in this way?
   
	Did you mean to square the above trig functions ?
 
    
------------------------------------------------------------------------    
  In answer to the general question, the answer is a definite NO.
Trying to do this would be equivalent to the invalid inference
pattern:

	p --> q		Invalid
	q
	_______
	p

For your example,
	p	sin^2( x ) + cos^2( x ) = 1
	q	1 = 1
 
  It may be that (for a given instance) the inference chain leading
from p to q is reversible, in which case the relation between p and q is a
logical equivalence and the desired result can be established:

	p <--> q	Valid
	q
	________
	p

  A (valid) proof that attempts to derive sin^2( x ) + cos^2( x ) = 1
(for example) from 1 = 1 will almost surely involve other theorems and lemmas.
(after all 1 = 1 is a rather weak result compared to some of the theorems I've
seen!) It is from these theorems and lemmas that the additional
hidden assumptions that establish the result will come.

  Just out of curiousity, do you have a derivation of the identity you
stated ( or some other readily recognisable trig identity ) that works
from 1 = 1 to the final result ?

			/R Best

as an addenda to .7:

  By the way, I have been trapped many times by this kind of fallacious
reasoning.  I think there is something very human and natural about trying to
use the  ( p -->q & q ) --> p pattern.  Often it appears in very subtle forms.

  Personally, I have to be very careful when doing proofs to be explicit
about writing down every assumption I use.  I think this is really the
hardest part of doing any proof: identifying the hidden contributors.
Over the years, I've accumulated a large number of mathematical 'facts' that
feel so natural that I apply them to jump from one step to another
sometimes without realising that I'm using them.

  Sadly, some of these 'facts' turn out to inapplicable to the exact
circumstances of the proof that I want to use them in because
I forgot some obscure prerequisite condition.  One of my favorite bugaboo side
conditions is ' .. function must be continuously differentiable throughout 
..'.

			/R Best

p.s. I would be interested in hearing about other subtle errors that math
noters have encountered in doing various proofs.

    re. .7
    
    Your question was: 
    
        Did I mean to square the trig function:
    
                1 - cos( x ) = sin( x ) 
    
    and my answer is:  
    
        NO, because if I did squared this it would have been true.
        The idea was tyo start with some False and show  1 = 1 from
        this false assumption.
    
    Kostas G.
    
    
    
                  

    re. .8
    
    Since you are interested in hearing about other subtle errors etc.
    take a look at the following notes: #479, 466, 450  in this notes
    file.
    
    Enjoy,
    
    Kostas G.