Choose from these sentences:

\exists k, j \in \mathbb{Z} such thatx = 2k+1 andy = 2j - Assume
x + y even impliesx andy have the same parity. - Without loss of generality, assume
x is odd andy is even - Presume the provided statement is false.
- Let
x andy 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 andy be opposite parity integers with even sum. x is even andy is odd orx is odd andy is even- Hence
x andy 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 thatx = 2k+1 andy = 2j - Assume
x + y even impliesx andy have the same parity. - Without loss of generality, assume
x is odd andy is even - Presume the provided statement is false.
- Let
x andy 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 andy be opposite parity integers with even sum. x is even andy is odd orx is odd andy is even- Hence
x andy have the same parity x + y = 2k+1 + 2j = 2(k+j) + 1

Your Proof: