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### Find a Counterexample

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Date: 8/29/96 at 11:5:51
From: Nina Morgan
Subject: Find a Counterexample

What is a counterexample of:

If a>b and b=5, then a >= 6.
```

```
Date: 8/29/96 at 11:26:45
From: Doctor Mike
Subject: Re: Find a Counterexample

Hi Nina,

I can give you a counterexample, sort of.  If a is any number
between 5 and 6, for instance 5.5 , then a is greater than 5
but a is not >= 6.  Another counterexample is 5.4321 and
another is 17/3 .

These both can be written as fractions. A couple of counterexamples
that can't are "the square root of 33" and "2*pi - 1" where pi is the
famous number 3.1415926535898......

Of course, there is an unlimited supply of others.

So why did I say sort of?  Because I wasn't completely sure of the
background of the problem.  If this statement was given in the
environment of whole numbers (also called natural numbers or counting
numbers or integers) then none of my fine counterexamples does the
job.  That is because the only integers greater than 5 are 6, 7, 8, 9,
10, etc.  It certainly is true about all of these that they are
greater than or equal to 6, so if we are talking about integers, there
is no counterexample.

I hope this helps.  Write back if you have other questions.

-Doctor Mike,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
Middle School Algebra

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