# quantifiers

• Jul 30th 2011, 06:45 AM
lukeheselden
quantifiers
Need help with
Express in quanti ers:
The sequence {A n}n is element of natural numbers, is strictly decreasing
Give the negation of as well.
I am thinking for all N ,there exists x1 < x2, then f(x1) > f(x2) , kinda of lost...
• Jul 30th 2011, 06:58 AM
Plato
Re: quantifiers
Quote:

Originally Posted by lukeheselden
Need help with
Express in quanti ers:
The sequence {A n}n is element of natural numbers, is strictly decreasing
Give the negation of as well.

There are of course several different ways one might do this.
Here is one. $\displaystyle \left( {\forall m \in \mathbb{N}} \right)\left( {\forall n \in \mathbb{N}} \right)\left( {m > n \to x_m < x_n } \right)$

The negation is, $\displaystyle \left( {\exists j \in \mathbb{N}} \right)\left( {\exists k \in \mathbb{N}} \right)\left[ {j > k \wedge x_j \geqslant x_k } \right]$
• Aug 1st 2011, 06:22 AM
pece
Re: quantifiers
You mixed up the inequalities : the sequence is strictly decreasing ; what you wrote is for a strictly increasing sequence.

I assume it's a typo, but i point it out for the original poster not to confuse everything.
• Aug 1st 2011, 06:34 AM
Plato
Re: quantifiers
Quote:

Originally Posted by pece
You mixed up the inequalities : the sequence is strictly decreasing ; what you wrote is for a strictly increasing sequence.