I am attempting to self-study proof techniques and your criticism of my following proof would be greatly appreciated. Feel free to nitpick minor/trivial things; that's how I'll learn!
Edit: I have appended a revised proof with the criticisms received so that other rookies can learn from my progress too.
Edit 2: And another, third time lucky?
Theorem: Let $n$ be an integer. If $3n$ is odd then $n$ is odd.
Let $P$ be the sentence "$3n$ is odd" and $Q$ be the sentence "$n$ is odd." We therefore have;
$P \rightarrow Q$
By the law of the contrapositive, we may obtain;
$\lnot Q \rightarrow \lnot P$
Which is translates to "If $n$ is not odd then $3n$ is not odd", or put another way; "If $n$ is even then $3n$ is even."
If an integer $n$ is even then there exists some integer $m$ such that;
$n = 2m$
By multiplying this by $3$ we may obtain;
$3n = 6m \equiv n = 2m$
It has therefore been shown that if $n$ is even so is $3n$ and that this is equivalent to showing that if $3n$ is odd then so is $n$.
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Attempt 2, taking into consideration previous criticism
Theorem: Let $n$ be an integer. If $3n$ is odd then $n$ is odd.
Let $P$ be the sentence "$3n$ is odd" and $Q$ be the sentence "$n$ is odd." We want to show that $P \rightarrow Q$ and that by taking contrapositives this is equivalent to showing $\lnot Q \rightarrow \lnot P$ which translates to;
"If $n$ is not odd then $3n$ is not odd", or put another way; "If $n$ is even then $3n$ is even."
If an integer $n$ is even then there exists some integer $m$ such that;
$n = 2m$
By multiplying this by $3$ we may obtain;
$3n = 6m$
This must still be even as an even integer multiplied by an odd integer produces an even one.
We must now show that $3n$ is even. If this is so, then $6m = 2k$ for some integer $k$.
As $3n=6m=2k$ we have;
$3n=2k$
It has therefore been shown that if $n$ is even so is $3n$ and that this is equivalent to showing that if $3n$ is odd then so is $n$.
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Attempt 3
Theorem: Let $n$ be an integer. If $3n$ is odd then $n$ is odd.
Let $P$ be the sentence "$3n$ is odd" and $Q$ be the sentence "$n$ is odd." We want to show that $P \rightarrow Q$ and that by taking contrapositives this is equivalent to showing $\lnot Q \rightarrow \lnot P$ which translates to;
"If $n$ is not odd then $3n$ is not odd", or put another way; "If $n$ is even then $3n$ is even."
If an integer $n$ is even then there exists some integer $m$ such that;
$n = 2m$
By multiplying this by $3$ we may obtain;
$3n = 6m$
Which can be rewritten as;
$3n = 2(3m)$
Thus we have shown that $3n$ is even, as it is equal to $2(3m)$ which, as an integer multiplied by 2, must be even.
It has therefore been shown that if $n$ is even so is $3n$ and that this is equivalent to showing that if $3n$ is odd then so is $n$.
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