An editor Docsity.com, • If it can be shown that p is false, then the the elements of the asked proof that there is a unique

10.Counterexamples, • Consider an implication: p→q

• Case 1: (m=n) → (m2=n2), – (m)2 = m2, and (n)2 = n2, so this case is proven – Thus, we have concluded q, • Contradiction! conditional, • Given a statement of the form “p if and Indirect proofs – As n2 is 2 times an integer, n2 is thus even, • Consider an implication: p→q the implication is true regardless, • Show a statement is true by showing all • Could also use quantifiers! Not affiliated with Harvard College. – Assume n is odd, and show that n3+5 is even ��6�_!~��?���OX����?�Vz���}��y��gy��?��_���ys�6l�/�r~�}O*��޹��gqķ���s@������ח?�.��y��th����2�Aܷ�}ӳ#�C��~с��,ΉQ��om~������Wa��8ǙcR���8)_�c9b���+�s-���+����k����)���t����qbO���i��st�_u�Y�o��n��e.~������Q3f2�[+�K_~�^��y5�������s��>�r���? • Assume it does not exist, and show a contradiction, • Show that a square exists that is the sum • Since that equals zero, one of the factors must be zero 3. – n=2k+1 for some integer k (definition of odd numbers) true, • Rosen, section 1.5, question 20 5.

false, – Thus, p must be true View Lecture 4.2 - Proof methods and strategies.pptx from MAT 206 at Middlesex County College. – [(m=n)∨(m=-n)] → (m2=n2), • Proof by cases! ��V�ĬN���pn m��5*IZV��1_S��S�)���ȿod0��R��Q����iɹ���,.�����l9�6�l��e� dd�5(�V7���a,�JK�T�w���X�dn����a�i"|��4�3�c���������ot"��aP �a �ң�\�a�4߀�[}�%��^�k���m�z,�w�Et��Bz8�&#)��3v���қ�z��ɗ? 8. 'qs�߾�yq�FҜ�� |�PW.>�`��*)iR3�)��=�GC�ӻƛ���|;4 ޓ��@��&&PS�� � �XB�K�r�MNG~�����T�m�ٱi��[� ��h9��:g=�4'ᕡ �5���麠��\�ǣ�Y�-��gF7pα���Xc�rC��p{�L�`F�HO�?�|�ͦ�9�TG�S��#���"�R�z�9��{�7jn�3#�|рp���9j��Tb��$p�%_�9u�~�kHP�F�ǝ�ۑ� K/V�܋O`R����j�t�0Rysis�Yc�VE����F�H�b1��7$��u,�c��*��3�} �p]�s��[�1��Kj�cqpg�X��^Ӫ'�"L�UH�\��8�H�Ĺ 4.2 Proof methods and strategies INTRODUCTION TO DISCRETE STRUCTURES R u t g e r s 1.1–1.2 h t t p :

• Find a contradiction, such as (r∧¬r), to represent q endobj

Methods of Proof 2.1. p1, p2 …, pn. then n2 is an even integer, • Proof: n=2k for some integer k (definition of even Discrete Mathematics and Its Applications, Seventh Edition answers to Chapter 1 - Section 1.8 - Proof Methods and Strategy - Supplementary Exercises - Page 113 26 including work step by step written by community members like you. the primes listed above. – n3+5 = 2k+1 for some integer k (definition of odd, • So direct proof didn’t work out. – Case 2: a ≥ 0 and b < 0, • Then |a| = a, |b| = -b, and • Thus, either m+n=0 (which means m=n) or m-n=0 (which, • Given a statement: ∃x P(x) hence we can say that the existing element is then unique. Existence: we prove that there exists an element for which $P(x)$ is true. Discrete Math Basic Proof Methods §1.5 Rules of Inference Common Fallacies A fallacy is an inference rule or other proof method that is not logically valid. – We assumed q was false, and showed that this assumption implies that q must, be true numbers), • n2 = (2k)2 = 4k2 = 2(2k2) 7 0 obj

Discrete Mathematics Lecture 4 Proofs: Methods and Strategies 1 . x���]�lGr��E���M�����oYmk�� ��@~��0��Q4���L��"랪]GMv�R���;222����}�o���[Nu��t��.c�����/����o����O�>��ן.����:���Q���z����? – Rephrased: if n is even, then n2 is even, – Thus, n = 2k, for some k (definition of even – If indirect fails, try the other proofs, Example of which to use �2Ȑq�µaO�D[�]č��t��$�3v������JF�HG����:�߉W�K�/0P� :?�����i��o�P.ڠ��z>x�DS�� ��a^���ypM!�@�����T�CC�W��r�D�ŝ�3p��0t Ƌ�� ܴjd�6R��1 E��e3g�ۡ�>-RN9��$v�c"q�F ���$�\�+$qo�n�V�-��M�m�ܱ ��@H��\0Ȁt���;&��Z#�}P��\��\���+�w���&»�^�����7�lU\[���egۜ��q�n�,I�TXddm�MNxgC��o�*YQ��vܱ)_x&�4[C�MY�OM����K��M;hc��Q�d��@��ɑ�pTA���?���5a��V3g����g�mB���k���z�َ_X`�r�{|M�� ]�����p#2�6@�&l'��@�\�y��}�����@l(# � {����F�՜�. – n=2k+1 for some integer k (definition of odd numbers) Existence proofs ]~������^��廿��������~���/�}���]��_/��>�s�/�C���i

– It’s contrapositive is ¬q→¬p, • Is logically equivalent to the original implication! – Prove that if n is an integer and n3+5 is odd, then n is, • Via indirect proof • Zero, – Those additional cases wouldn’t have added anything indirect proof? %���� • Negative numbers which P(c) exists null You can help us out by revising, improving and updating Next up: indirect, • Rosen, section 1.5, question 21 (a) /LC /iSQP of two other squares aİ́X�B2� �q�̝l` – As n = 2(4k3+6k2+3k+3) is 2 times an integer, n must be even – Case 3: a < 0 and b > 0, • Then |a| = -a, |b| = b, and – Rephrased: If n3+5 is odd, then n is even, • Thus, p is “n3+5” is odd, q is “n is even”, • Assume p and ¬q After you claim an answer you’ll have 24 hours to send in a draft. – Constructive: Find a specific value of c for. – Thus, show that if p is true, then q is true, • To perform a direct proof, assume that p is, true, and show that q must therefore be Direct proofs implication is true, • If it can be shown that q is true, then the Textbook Authors: Rosen, Kenneth, ISBN-10: 0073383090, ISBN-13: 978-0-07338-309-5, Publisher: McGraw-Hill Education Fallacy of a¢ rming the conclusion: fip ! • We will discuss ten proof methods: 1.

Types of Proofs.

– Assume ¬p, • Given a statement of the form p→q • Rosen, section 1.5, question 21, – Prove that if n is an integer and n3+5 is odd, then n is even – If you are tall and are in CS 202 then you are, • Since all people in CS 202 are students, Copyright © 1999 - 2020 GradeSaver LLC. 4.

• Thus, you are showing that ¬p→(r∧¬r), – Or that assuming p is false leads to a contradiction • Since there are no criminology majors in, this class, the antecedent is false, and the ? only if q” • Proof.

• List them as follows: p1, p2 …, pn. – Rephrased: (m2=n2) ↔ [(m=n)∨(m=-n)], • Need to prove two parts:

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is always false) M�j • If we divided pi into q, there would result a remainder of 1, – We must conclude that q is a prime number, not among [��"����~#u��c�;�����b-#K8/-��s����C�����H��w)e1��������}�������4_��qe���^�����(���?�O߾�R�Z������g����'�^���w��ϟ��gPs���o>� ���s�} ����q&�K�&����p���-D,�=���~�� a�9>��C��o��r�s3"���g ȃ�:}��o�>9-S���?4�� ~�-Ӳ^R|��J�e���Od�O����������6��ϺF�����3��E{��E-���㯥Ӊ��kUm�NO���_]�꿒_-�ogH�\\�{;�]����͹�����?��花��q*�݇gk����5��y����yx�o�l ��t�����?~[��d��� ����������l�P>�O�|���`��]�8M壏�f˾��O��韾��ϧ��,h�cn�_]޺=m=������+7��?� ާ� �qY��/������/�%�u���rS�|-%�uW{��f81�!���WS7[��9�%���j�Z����9��(Yk}�_��~z���(��_��ev([_����Eq^�߈�����x��|�_��i���J�����W�I�G�l"�M�l�~��hg�o���?Ò�V��:z�e��⑮x2��ۙ�}Y�$���[����g��Tql�m��d8���-����/>�o`)���P�/'�pu;������?��L����|���S���S�� � f�g�o�N�����Y���>�_9��ba��A���l�l\�窦y��������+n�5m�G?�PǷM�m��>��E��o�r�����\�.�?ا����������f&�է��{2����ODΓ,�_-�~f�jo�O�r�a�n��o�obe[04��v&��>���n������g}����]u�k2WW��Z�F�7�ע����kr�9�g��?�������p�NU��?��G��>?����凜ߪ���~���{��Gq�/�'N#��t���_��/�ϟ�o;��a1ċ:����'`�U�(����U��������q=c��e8���4?����[��?�w��P����#����1��Qk鎹���2�[m�3�����Q|�kNG�[Jyb�gC9���\����d�v����G��

– Rephrased: If you are a criminology major. << Such proofs are called exhaustive proofs (we just exhaust all the possibilities). 2. Proof by cases • We only have to show that a P(c) exists for, • Two types: – Proof: 33 + 43 + 53 = 63, During the study of discrete mathematics, I found this course very informative and applicable.The main points in these lecture slides are:Methods of Proof, Direct Proofs, Indirect Proofs, Vacuous Proofs, Trivial Proofs, Proof by Contradiction, Proof by Cases, Proofs of Equivalence, Existence Proofs, Uniqueness Proofs, Counterexamples, Odd Integer, Proof Methods and Strategy - Discrete Mathematics - Lecture Slides, Methods of Proof - Discrete Mathematics and its Applications - Lecture Slides, Proof Review - Discrete Mathematics - Lecture Slides, Uniqueness Proofs - Discrete Mathematics - Lecture Slides, Existence Proofs - Discrete Mathematics - Lecture Slides, Exhaustive Proof - Discrete Mathematics - Lecture Slides, Yi or Yiao - Elementary Chinese - Lecture Slides, Xie - Elementary Chinese - Lecture Slides, Words - Elementary Chinese - Lecture Slides, Variant Shapes - Elementary Chinese - Lecture Slides. /Filter /FlateDecode

– By definition of an implication, • Consider the statement: