Negating a Quantifier

Date: 03/20/2010 at 20:18:02
From: Marianne
Subject: Converse, Inverse, contra positive

I have to write the converse, inverse, and contrapositive of this
conditional statement:

   If a triangle is isosceles, then it has at least
   two congruent sides.

I know that

                p: a triangle is isosceles
                q: the triangle has at least two congruent sides
        statement: if p then q 
         converse: if q then p 
          inverse: if not p then not q 
   contrapositive: if not q then not p 

So

   Converse: If a triangle has at least two congruent sides, then the
   triangle is isosceles.

But what is the negation of "at least two"? Is it "none"? or "at most
two," as in

   Inverse: If a triangle is not isosceles, then it has at most two
   congruent sides.

   Contrapositive: If a triangle has at most two congruent sides, then
   it's not an isosceles.

Date: 03/20/2010 at 22:45:17
From: Doctor Peterson
Subject: Re: Converse, Inverse, contra positive

Hi, Marianne.

Just consider the cases. It either has none, or two, or three:

   Number of
congruent sides    At least 2?    At most 2?    None?
---------------    -----------    ----------    -----
       0                F             T           T
       2                T             T           F
       3                T             F           F

Which column is the negation of "at least 2"? The one that switches 
every falsehood to a truth and every truth to a falsehood -- the one 
titled "None?"

There are other ways to come up with this fact. You might write it as
an inequality.

   N is at least 2
   
means

   N >= 2
   
The negation of that is

   N < 2
   
which in turn means 0 or 1. Since a triangle can't have "one congruent
side," that really means none.

Or, by the same reasoning as the last sentence, if a triangle does NOT
have at least two congruent sides, then it has none. Saying it has two
congruent sides means it has a PAIR of congruent sides, and if it 
doesn't have at least one pair, then it has none.

So your inverse and contrapositive are wrong. (You could also directly
check them: the contrapositive should be true, but if you make a 
triangle that has at most two congruent sides, it might have two, and
WILL be isosceles.)

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/ http://mathforum.org/dr.math/

Date: 03/21/2010 at 00:32:31
From: Marianne
Subject: Thank you (Converse, Inverse, contra positive)

Thanks so very much!!!