What would be the truth value for the following two quantifiers if n and m are both integers? I have trouble proving each of these statements. I appreciate any help you can provide!
a) $\forall n\; \exists m\; (n^2 < m)$
b) $\exists n\; \forall m\; (n < m^2)$
On
a) $\forall n\; \exists m\; (n^2 < m)$
For all integers $n$ there is an integer $m$ such that $m$ is strictly greater than the square of $n$.
So, if you select any arbitrary integer, can you always find another integer that is strictly larger than its square? How?
b) $\exists n\; \forall m\; (n < m^2)$
There is some integer $n$ such it is strictly less than the square of any integer $m$.
So, can you find an example of an integer that is strictly less than all squares of any integer? Why?
The first is true. Just let $m = n^2 + 1$.
The second is also true. Let $n = -1$. For any $m$, $m^2 \geq 0$, and $n < 0$.