Let n be an integer Prove that 3n 11 is odd if and only if Let n be an integer Prove that n 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 2n 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 10, 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

Question

Let n be an integer  Prove that 3n 11 is odd if and only if Let n be an integer  Prove that n 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 2n 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 10, 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

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Question

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

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Discrete and Combinatorial Mathematics An Applied Introduction

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