Consider the following expression about natural number: \forall n\exists m: m^{2}=n .

Question

$$ \forall n\exists m: m^{2}=n $$ How do I interpret this? This is part of my assignment question but I cant carry on without understanding this.

Answer 1

If $n$ is a natural number, the expression $n=m^2$ for some $n$ tells us that every natural number can be expressed as a product of a real number $m$ to itself. To see this note that:

$0=0^2$

$1=1^2$

$2=(\sqrt{2})^2$ Etc.

Bear in mind that $m$ is not necessary an integer.

It also works even if $n$ is a real number.

Answer 2

To make your statement complete (where "$\mathbb{N}$" indicates the natural numbers):

$\forall n \in \mathbb{N} \ \exists m \in \mathbb{N} \mid m^{2}=n $.

This is false, because $2$ is a counterexample.

To test your understanding:

Is this true or false?

$\forall m \in \mathbb{N} \ \exists n \in \mathbb{N} \mid m^{2}=n $.