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!!!
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