Question

Write the negation of this statement, avoiding use of the word "not"
"For all real numbers x, there exists a positive integer N such that \(N-1\leq x^2\leq N\) "
Please, help

Answer 1

we have to use
https://en.wikipedia.org/wiki/There_exists#Negation

Answer 2

Thanks a ton. :)

Answer 3

Ok, check my work please
I symbolize the statement as
\[\forall x \in R, \exists~~ N >0~~ | N-1\leq x^2 \leq N\]
\[\neg(\forall x \in R, \exists~~ N >0~~ | N-1\leq x^2 \leq N)\\\exists x \in R, \forall N> 0 ~~ | N-1 \geq x^2 \cup x^2\geq N\]
Verbalize it: "There exists a real number x, for all positive N such that x^2 is lesser or equal N-1, or x^2 greater or equal N"
Am I right?? Please check.

Answer 4

there exist a real number x , such that for all possitive N
N-1>x^2
or
x^2 >N

so your correct ^^

Answer 5

Thank you @ikram I made mistake at \(\geq\) . It should be > only. hihiihi...