Just refine the "choose" sentence statement with its order of the proof table by re-arranging the statement. Read the statement and fill the proof with number list corresponding to the proof statement only;

(1) Order and refine seven of the following sentences so that they form a logical proof by contrapositive of... Claim: "If 9x+19 is even, then x is odd." I say "refine" because some of the conventional connecting words, such as "therefore" are left out. Make it flow. Also, some details are missing; for example, we are left to wonder, "what type of number is ?" Fix this. Choose from these sentences: Your Proof: 1. 9x + 19 is even Number from list The statement. 2. Suppose 9x + 19 is not even. 3. 9x + 19 = 2j + 1 for some integer / 4. 9x + 19 = 9(2k) + 19 = 18# + 18 + 1 5. x is even. 6. Then a miracle occurred. 7. 9x + 19 is odd 8. Suppose x is an integer that is not odd. 3k such that x = 2k 10. 9x + 19 = 2(9k +9) + 1(2) Order and refine eight of the following sentences so that they form a logical proof by contrapositive of... Claim: "If the sum of two integers is even, then they have the same parity." Again, I say "refine" because some of the conventional connecting words, such as "therefore" are left out. Make it flow. Also, some details are missing; for example, we are left to wonder, "what exactly is this number s?" Fix this. Choose from these sentences: Your Proof: 1. Assume x + y even implies x and y have the Number from list The statement. same parity. 2. x + y is odd 3. 3k, / E Z such that x = 2k and y = 2/ + 1 4 Without loss of generality assume x is even and y' is odd. Either x is odd and y is even or x is even and y is 5. odd. 6. Suppose x and y are integers with opposite parity. 7. x+ y = 2k +2j+1 8. Therefore, x + y is even = x and y have the same parity. 9. 3s such that x + y = 2s + 1