How to write and proof of the following statement?
If n is positive number and not negative - than 5n + 3 is even number, only if n is odd number.
Here is my trial:
If p is positive number
$\forall n (P(n) \land \neg P(n) \rightarrow Q(n)) \rightarrow R(n)$
Well, you have the nesting correct. You should also embrace the entire scope of the quantifier, since implication have precedence.
If $P(n)$ means "$n$ is positive", then $\neg P(n)$ means "$n$ is not positive." You need an additional predicate.
$\forall n\left({\left({\left({P(n)\wedge \neg S(n)}\right)\to Q(n)}\right)\to R(n)}\right)$ where $P(n):= (n>0) \\ S(n):=(n<0) \\ Q(n):= (2\mid (5n+3)) \\ R(n):= (2\nmid n)$
However, $P(n)\wedge\neg S(n)$ is redundant. You really just need to specify $P(n)$
$$\forall n\left(\raise{1pt}{\left({P(n)\to Q(n)}\right)\to R(n)}\right)\\ \forall n\left({\left({(n>0)\to (2\mid (5n+3))}\right)\to (2\nmid n)}\right)$$
To evaluate the truth of this statement either: