Proofs on tplat

If $n$ is even, then $n^2$ is even

Mon, 01 Jan 0001 00:00:00 +0000

Claim:

If $n$ is even, then $n^2$ is even.

Proof:

Assume $n$ is even. Then, by definition, there exists an integer $k$ such that:

$$ n=2k $$

Then:

$$ n^2=(2k)^2 =4k^2 =2(2k)^2 $$

Since $2k^2$ is an integer, $n^2$ is divisible by $2$. Therefore, by direct example, $n^2$ is even.

Claim:

If $n^2$ is even, then $n$ is even.

Proof:

Assume $n$ is odd. Then, by definition, there exists integer $k$ such that:

Mon, 01 Jan 0001 00:00:00 +0000

Claim:

If $n$ is odd, then $n^2$ is odd.

Proof:

Assume $n$ is odd. Then, by definition, there exists an integer $k$ such that:

$$ n=2k+1 $$

Then:

$$ n^2=(2k+1)^2 =(2k+1)(2k+1) =4k^2+4k+1 =2(2k^2+2k)+1 $$

Let $m=2k^2+2k$, which is an integer. Then:

$$ n^2=2m+1 $$

Therefore, by direct example, $n^2$ is odd.

Claim:

If $n^2$ is odd, then $n$ is odd.

Proof:

Consider the contrapositive; if $n$ is odd, then $n^2$ is odd. We have already shown this to be true. Therefore, by contrapositive, $n$ is odd.

Mon, 01 Jan 0001 00:00:00 +0000

Claim:

If $x^2>0$, then $x>0$

Proof:

Let $x=-2$

Then $x^2=4>0$, but $x>0$ is false.

Therefore, by counterexample, the statement is false.