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Order 8 of the following sentences so that they form a logical proof by contradiction of the statement:

If the sum of two integers is even then they have the same parity.

Choose from these sentences:

$\exists k, j \in \mathbb{Z}$ such that $x = 2k+1$ and $y = 2j$
Assume $x + y$ even implies $x$ and $y$ have the same parity.
Without loss of generality, assume $x$ is odd and $y$ is even
Presume the provided statement is false.
Let $x$ and $y$ be integers with the same parity but with an odd sum.
Parity is not knowable without a paring knife
$x + y$ is odd
Let $x$ and $y$ be opposite parity integers with even sum.
$x$ is even and $y$ is odd or $x$ is odd and $y$ is even
Hence $x$ and $y$ have the same parity
$x + y = 2k+1 + 2j = 2(k+j) + 1$

Your Proof:

Choose from these sentences:

$\exists k, j \in \mathbb{Z}$ such that $x = 2k+1$ and $y = 2j$
Assume $x + y$ even implies $x$ and $y$ have the same parity.
Without loss of generality, assume $x$ is odd and $y$ is even
Presume the provided statement is false.
Let $x$ and $y$ be integers with the same parity but with an odd sum.
Parity is not knowable without a paring knife
$x + y$ is odd
Let $x$ and $y$ be opposite parity integers with even sum.
$x$ is even and $y$ is odd or $x$ is odd and $y$ is even
Hence $x$ and $y$ have the same parity
$x + y = 2k+1 + 2j = 2(k+j) + 1$

Your Proof: