So I am given that n is a square iff there exists a k $\epsilon$ $\mathbb{Z}$ such that n = ${k^2}$. So our domain of discourse are the integers, we can define predicate Square(n) := $\exists$ k (n = ${k^2}$).
I am asked to write the following claim in predicate logic: If integers n and m are squares, then nm is a square.
I wrote:
$\exists$ n $\exists$ m (Square(n) $\wedge$ Square(m) $\rightarrow$ Square(n*m))
Is this correct? I am unsure if it is correct since the question also tells me to be careful, but I don't see what I should be careful about since it seems somewhat straightforward.
It is not correct but really close, just pay attention to the quantifiers. Notice that it is "If integers $n$ and $m$ are squares, then $nm$ is a square". The formula inside is correct though.