# Thread: Finding the Converse & Contrapositive of a Statement

1. ## Finding the Converse & Contrapositive of a Statement

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."

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?

2. 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.
$\left( {\exists x \in \mathbb{R}^ + } \right)\left( {\forall n \in \mathbb{N}^ + } \right)\left[ {\frac{1}{n} \geqslant x} \right]$

3. 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!

4. 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!

5. 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"?

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

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

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

8. 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.

9. That helps.....Thank You!