Finding the Converse & Contrapositive of a Statement

**dolphinlover** · October 11th 2008, 10:22 AM

Hi all,

I need to find the converse & contrapositive of the following statement:

"For all positive real numbers x, there exists an integer n such that
1/n < x."

Thanx in advance for your help!
Oh.... and I found the negation of this stmt. to be:

There is a positive real number x such that for all n, if n is not an integer, then 1/n is greater than or equal to x (I don't know how to use latex).....is this right?

**Plato** · October 11th 2008, 11:26 AM

Originally Posted by dolphinlover

There is a positive real number x such that for all n, if n is not an integer, then 1/n is greater than or equal to x (I don't know how to use latex).....is this right?

Here is the negation.
$\left( {\exists x \in \mathbb{R}^ + } \right)\left( {\forall n \in \mathbb{N}^ + } \right)\left[ {\frac{1}{n} \geqslant x} \right]$

**dolphinlover** · October 11th 2008, 11:50 AM

That is what I have...can you help with the converse and contrapositive?(symbols are fine) I am stuck on that part & I have several more problems that are similar to this one.

Thank You!

**Plato** · October 11th 2008, 11:58 AM

Originally Posted by dolphinlover

"For all positive real numbers x, there exists an integer n such that
1/n < x."
There is a positive real number x such that for all n, if n is not an integer, then 1/n is greater than or equal to x (I don't know how to use latex).....is this right?

[/QUOTE]

Originally Posted by dolphinlover

That is what I have...

No it is not what you had.
Look at the text in “red”. That is wrong!

**dolphinlover** · October 11th 2008, 12:12 PM

Hi again Plato,

So your negation in words reads..."There is a positive real number x, s.t. for all positive natural numbers n, 1/n is greater than or equal to x"?

**Plato** · October 11th 2008, 12:32 PM

Originally Posted by dolphinlover

So your negation in words reads..."There is a positive real number x, s.t. for all positive natural numbers n, 1/n is greater than or equal to x"?

Correct: $\neg \left( {\forall x} \right)\left( {\exists y} \right)\left[ {P(x,y)} \right] \equiv \left( {\exists x} \right)\left( {\forall y} \right)\left[ {\neg P(x,y)} \right]$

$P \Rightarrow Q\,\mbox{ converse } \,Q \Rightarrow P$
$P \Rightarrow Q\,\mbox{ Contrapositive } \,\neg Q \Rightarrow \neg P$

**dolphinlover** · October 11th 2008, 12:45 PM

Symbolically, I understand the converse and contrapositive. The problem I'm having is with the Quantifiers being part of the implication.

**Plato** · October 11th 2008, 12:54 PM

Originally Posted by dolphinlover

The problem I'm having is with the Quantifiers being part of the implication.

Well translate it into standard English without quantifiers.
If x is a positive real number then there is a positive integer, n, such that (1/n)<x.

Now you have P & Q.

**dolphinlover** · October 11th 2008, 01:00 PM

That helps.....Thank You!

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