How would I got about proving a number is odd using propositional logic?
An odd number is $2k+1$ essentially the result of an even number +1 which gives you an odd number. For example $5$ is the result of $4+1$, $4$ being the odd number this is how I went about showing how a number is odd, but how would I do this using propositional logic? Would the way I used above to show that a number is odd be an informal or formal proof?
In the realm of integers, one may define "$n$ is odd" as $$ \neg\exists k\colon n=2\cdot k$$ or as $$ \neg\exists k\colon n=k+ k$$ while some authors may indeed define it as $$ \exists k\colon n=2\cdot k+1.$$ Which definition an author uses is perhaps somewhat up to taste. Of course, it should be noted that all three definitions are (provably) equivalent (in the realm of $\Bbb Z$ as well as $\Bbb Z_{\ge 0}$, but not in $\Bbb Z_{>0}$). Of course, the last variant is easier to use in a constructive proof that some specific $n$ is odd (namely, by exhibiting some $k$ with $n=2k+1$).