Let n be an integer. Prove that 3n-11 is odd if and only if Let n...

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Let n be an integer. Prove that 3n-11 is odd if and only if Let n be an integer. Prove that n³ is even if and only if n is even. Let n be an integer. Prove that n is odd if and only if n is odd. Let n be an integer. Prove that 2n²-n-1=0 if and only if 3n2-n-2=0. Give a proof of n is even. Let ne Z. Then n-3 is even if and only if n +4 is odd. using (a) two direct proofs. (b) one direct proof and one proof by contrapositive. (c) two proofs by contrapositive. Let z be a real number. Prove that if +5z+1≤0, then <0. Letr and y be integers. Prove that if z+y29, then either z≥ 5 or y ≥ 5. Letz and y be integers. Prove that if 2r +3y2 1, then z21 or y 21. Let a, b and m be integers. Prove that if 2a +362 12m+ 1, then a 23m +1 or b2 2m +1. Let a,b and e be nonnegative integers. Prove that if a +26+3c25, then a ≥ 3, 6 ≥ 2 or c2 1. Consider the following: Result Let ne 2. Then 5n+7 is even only if n is odd. Next, consider the following: Proof. Assume first that n is an odd integer. Then n=2a+1 for some integer a. Therefore, 5n+7=5(2a+1)+7=10a+5+7=10a+12=2(5a + 6). Since 5a +6 is an integer, 5n+7 is even. Next, assume that n is an even integer. Then n = 2b for some integer b. Hence 5n+7= 5(2b)+7=106+7=2(56+3)+1. Since 56 +3 is an integer, 5n+7 is odd. What is wrong with the preceding proof? Let n be an integer. Prove that 3n-11 is odd if and only if Let n be an integer. Prove that n³ is even if and only if n is even. Let n be an integer. Prove that n is odd if and only if n is odd. Let n be an integer. Prove that 2n²-n-1=0 if and only if 3n2-n-2=0. Give a proof of n is even. Let ne Z. Then n-3 is even if and only if n +4 is odd. using (a) two direct proofs. (b) one direct proof and one proof by contrapositive. (c) two proofs by contrapositive. Let z be a real number. Prove that if +5z+1≤0, then <0. Letr and y be integers. Prove that if z+y29, then either z≥ 5 or y ≥ 5. Letz and y be integers. Prove that if 2r +3y2 1, then z21 or y 21. Let a, b and m be integers. Prove that if 2a +362 12m+ 1, then a 23m +1 or b2 2m +1. Let a,b and e be nonnegative integers. Prove that if a +26+3c25, then a ≥ 3, 6 ≥ 2 or c2 1. Consider the following: Result Let ne 2. Then 5n+7 is even only if n is odd. Next, consider the following: Proof. Assume first that n is an odd integer. Then n=2a+1 for some integer a. Therefore, 5n+7=5(2a+1)+7=10a+5+7=10a+12=2(5a + 6). Since 5a +6 is an integer, 5n+7 is even. Next, assume that n is an even integer. Then n = 2b for some integer b. Hence 5n+7= 5(2b)+7=106+7=2(56+3)+1. Since 56 +3 is an integer, 5n+7 is odd. What is wrong with the preceding proof?