Problem #1

Question: say whether the following is true or false and support your answer by a proof.
$$(\exists m \in \mathcal{N})(\exists n \in \mathcal{N})(3m + 5n =12)$$

Answer: it's false.

Claim: $(\exists m \in \mathcal{N})(\exists n \in \mathcal{N})(3m+5n=12)$
Proof: suppose it's true and there is natural numbers $m$ and $n$ such that $3m+5n=12$. Let's find that numbers. It's clear that $(\forall m \gt 2)(\forall n)(3m+5n \gt 12)$ and $(\forall n \gt 1)(\forall m)(3m+5n \gt 12)$, so we should consider $1\le m \le2$ and $n=1$. There is only two possible variants in this case:
$$\begin{equation} 3\cdot1 + 5\cdot1 = 8 \end{equation}$$
$$\begin{equation} 3\cdot2 + 5\cdot1 = 11 \end{equation}$$
But as we see $3m+5n\neq12$ for all possible natural numbers $m$ and $n$ and thus we can conclude that the claim is false. $\blacksquare$