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Proofs that prove a theorem by exhausting all the posibilities are called exhaustive proofs i.e., the theorem can be proved using relatively small number of examples. �GB�6��ث��|���rU�;�a��$�"-l�(\DE��H�ک6�'� ���%���:���ɨ�0�;�_�H� V�V�.In�H������&c"�;:�稢F�BK�;��S.p_�\��p���y3�>��9�+@B�B�QIU�b����,k6! �l+�M��E���-���"i�����X����P+�,�} N�x��m�a��,��̵�w�F�;",��;��E���X�۶c�H@̈́n��«��"��%���@�|�L+*,N�Ж|H�%�� %PDF-1.5 Whereas, in calculus, it is continuous functions of a real variable that are important, such functions are of relatively little interest in discrete mathematics. s���1G��5�C!ڶ���j� �Æ��(�yt_`�;|C9�BxO����VѱBT b ֱ��wnj�u����n�) ���!C��]>�6�ӱE�,D=-�����g_���� ���H�D����/ ��x��nZ��FT"�E�?�x���QO��� 95��ע�f�' �iS��x�2Ơu�x�F�~ ���e�7�ȼ��:Xm�1.є�N4ͱ��޲�Ê�:2�x��QO�܈�������-s���_�V�m�D��# '���7,>�T�>^? Chapter 1 The Foundations: Logic and Proofs The word \discrete" means separate or distinct. Revisiting the �N`M����(�(N \3D��{��>/��-Z�ϋ��2x�f�)s�h ޓGd�����ߙ� Section Summary Valid Arguments Inference Rules for Propositional Logic Using Rules of Inference to Build Arguments Rules of Inference for Quantified Statements Building Arguments for Quantified Statements. <> stream 6 0 obj m�'b����-��[��{��:vT����6i The Foundations: Logic and Proof, Sets, and Functions his chapter reviews the foundations of discrete mathematics. ٩�� k&�� ;�eIޖ塕�[�N��8����'�}�/�������@ �@ �@ �Xb�)=M ,19�0k�LI�z��B��U�z��߼j��,@j]�F�l�Cc̈́�����5��Z� �Kb?���ঔ�����Z").v}�\hd�Nł�����M��jDD�vR[$̣{X�Agh���C���T#���9.L� <>>> 0 Logic and Proofs.pdf from AA 1THE FOUNDATIONS: LOGIC and PROOFS Foundations of Mathematics Oreste M. Ortega, Jr. Leyte Normal University Foundations of … In math, CS, and other disciplines, informal proofs which are generally shorter, are generally used. Functions Definition: Let A and B be nonempty sets.A function f from A to B, denoted f: A →B is an assignment of each element of A to exactly one element of B.We write f(a) = bif bis the unique element of B assigned by the function f to the element a of A. <> C$G�Tr�Ύ�� �K\y鶋�c������ ���'(�a�����4�l�A`�����or ���y�*��s5����' /v�ԮO���:��wF|�. 3 0 obj Functions are sometimes called mappings or transformations. Rules of Inference Section 1.6. �j���p��T��C��)i$#*�Tx3�{���R��z=z4���Ϡ������1�� �-��t0ҭ��r��h�D��!+�|����k� endstream TUQ����y��Mg,��`}��ś���k�ׯ���}�N@_��{G���s찚B�'�T�.3��po\-���4���-��sт@ �[ה���K��{�u��Xq�3��IE#�uԈ���G�� �+Av�Rb � ���o�w o�D`�[�Û4- �� �K�NSj�� ڴSk�ro�������EH�~�E�V"��\ %DzD�v��1�[�CЎ ��T'��1��f�=z0��1�> M�����H�߆�\�s��=x�vI�ȹh~����1p�^݃fS&��Q7G=�>^bʥ,�a�R�f���-����R���t��ҷ�Z���O�i�dæ���� endobj <> Propositional Logic • The Language of Propositions • Applications • Logical Equivalences Predicate Logic • The Language of Quantifiers • Logical Equivalences • Nested Quantifiers Proofs • Rules of Inference • Proof Methods • Proof Strategy ��ͤ@��Lsy$�u �M��#�^��f���T��,I$h.���2�&������՞�~����h1�k*)6�$���uV��k����l)�$� � e�=�*P� ���cZ"H$8B�)Mv���g�`�3�U�D�?�j�ٰČI�F��V.��� endobj endobj Z� ����+dq`¸�-62�LZPZ1�"�G�PR!IQ�B��$�.PK�����gm1û4�����O��# J2�&�x�i �u~^O�؅�E�B��i�AO Section Summary Valid Arguments Inference Rules for Propositional Logic Using Rules of Inference to Build Arguments Rules of Inference for Quantified Statements Building Arguments for Quantified Statements. <> %PDF-1.5 6���s�N���H�9Abо��T1w@�;��UВ�ȱ,hU7� ���;��}���kƉ�#̣4�N�=K�� ��H^%g}K��/��Ս;�0���c[���7�o��_�����F��cd�fS{�;�͐æ���=�ʜ-��OEcvE2�c�fΪ]��%ٱ��9o_�ļx Y#��/\C�����QO~������~�]@0��=ė���=®����������[� 4 0 obj �~ox��������`έ຅����>�{pupTI������O�s/���2�O-\�\ry��+!�I��u�QZ�4ʨ�3�1B%,`~�F Ն�_�! Ê�:bX���޼Y���-Vj�=���:sl���gD=���{� [��q��k'� ͵j��Ш�p�~l���)V���&����z�w�z��q��F�H@��ئ٦���8߅���KJ5��e�r�s�|�E���_L1w�%��� �� ���Pc�MQh| #������U�᭑`���:~\Գ�ÃڲZ �͐ބ}����9дP���>5('f������+( ��}�lW2��n3�w� �^�Q�s������p}��75�hHY��珡vz��H�^�u�Z��l�~hi�/}��xN2nZ���spZOR���^���c�,�}L���I����C_�Z�� ��&�4թ�a���DÉ+�F=c?jؗ���O?�z���V ��[7�G眹j��c�3w����F2ԉ�Ś�>�g]�����z1ef��w��y�� ?�6Qu��cdS�m�x���>琌�!�SF}�؃f2�U� o����6�S�e��W��O�P/�a�8ՕN�]:�7n��p�~�v��2o�B͗Meh8s�ު�j�^��z�.5_x�l��^���g�>�}:m�*G��\z�oP��5_�ơw������g�c�Lu�[��r��V+�1�D����ub�I��$ �ƖaC�ZH7��I�����Ӝ���3���7k���@ �@ ��%6o��2d�c�lj�jz�6zǯ j�Z��E���ȼ�X�� �w@ �L&��/^��ַ�_�wF&e?��l�.��X�;~�DBh&t=�^��ѷ���[Y�i�(��8P[�����]VRGx, <>/XObject<>/Font<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 720 540] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> 2 0 obj • Steps may be skipped. �8p�9RrNr0�C����l8�}1�*���s+�n�����O���_4*�W����=���O��ja�:�����^ �Lr|h�C���PD=�)�������u.8�����絥Q�%Q�Lk�I�P��!�� �u��S�� g�����3�o��;��q�p,k��uH!n�5�]���6�|�E�����_I�?����%���m?�f�˧��o%�D��b�?G��'5Km'�. 1 The Foundations: Logic and Proofs 1.1 Propositional Logic 1. a proposition is a declarative sentence that is either true (T) or false (F), but not both. endobj stream x��Vێ�6}7��G2�i�IA�dou� 6��Ԇ,�:3s����͛�����۷��� ��%� U G�}1x㦠�� ���+f�� � �����\87`f=B��sh��ꣅl���}Zb ꔫ�:E��-�z7ef�YR�#ӹ3�Ԍ-|`ԽVQ�X�� E�N�N��.�n�R���_��{z Z�Î�7��`` ���d������3�v�u�8��?�n�1��_d۾�h��U���ֱ�E�\�jo����B�����j����]xOL�}Ρ�H��-��Ĺ7���J�J��1��E::���C��8�r�xj,����P{߹SP{���eK�^�a�~\��['�-�7���>} %���� Proofs of Mathematical Statements A proof is a valid argument that establishes the truth of a statement. endobj 5 0 obj x���������o �$��f3�gٻtQƧe�R�M�PJ��R �Z�V�H8�%�h����}rOg�0C���I���ҳ�K���`�1͔^7���B��Gg�g�d�jP5�{wt��;����`�2�}�Dg��9����ᄈߥ��E���w�p�6mƯ3`i���+SF����#p 9X�-Wj$��!�����.x�4(��0{3�,9�h�Z-Vx���0��j�}���|���F���G�珡�`�2 sR׸��W+Mˠ��pLiu����M! 2 0 obj The Foundations: Logic and Proofs Chapter 1, Part III: Proofs. ��,��_hW��4��QI]� 1��HGf � &G?q4)6c 7{H�T �:"��E`c� endobj [z��}�$���" ڷW�'u�=X���1,)�6e�FIa�)����=z�0%�n6�e�gM��;�ao�S��NS��7�gZG�r���[p��{�}Yę�pEE죇�'�M���b*�@���(�P�� ����) ~,6�� A B C Students Grades D F Kathy Scott Sandeep Patel <> Example: Prove that ( n + 1) 3 ≥ 3 n if n is a positive integer with n ≤ 4 Summary Valid Arguments and Rules of Inference Proof Methods Proof Strategies. endobj 4 0 obj • More than one rule of inference are often used in a step. The Foundations: Logic and Proofs Chapter 1, Part III: Proofs. Mathematicians view it as the opposite of \continuous." 1 0 obj 1 0 obj <> ���Г�T��,I$h.���6�&�������^�~��t��[*�����0�m8�Ag&�K�b#ˤ��.�y$���2d/;y��Zb�n6k^�b��ldY�n?ӖG�d�ML8Ϝi �U~^M����:}�-�"E=�Hft3�M��U� C� Kh8�/�@�T#�s�@���W�m�z�~`9��ܦ6�!�:s�������k���JKߕ�Xj:����p-[���u¸b>L'�wZ�W�x��?Ǒ�� �c��S�����s�rcl"� �0p̚N����0���g�Tg��۷�"��J��˰� stream Three important topics are covered: logic, sets, and functions. 9V%��cD��%�{�ON�� �8�� �,�4��-[���ѽ ���$�����ݔQu��7aN 3 0 obj %���� Rules of Inference Section 1.6. @{�����������Rf6�g�2��=}B���lU

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