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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�%��� 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. <>/XObject<>/Font<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 720 540] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> Whereas, in calculus, it is continuous functions of a real variable that are important, such functions are of relatively little interest in discrete mathematics. 1 0 obj stream ��ͤ@��Lsy\$�u �M��#�^��f���T��,I\$h.���2�&������՞�~����h1�k*)6�\$���uV��k����l)�\$� � ��,��_hW��4��QI]� 1��HGf � &G?q4)6c 7{H�T �:"��E`c� endobj 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���>} Functions are sometimes called mappings or transformations. 4 0 obj A B C Students Grades D F Kathy Scott Sandeep Patel �~ox��������`έ຅����>�{pupTI������O�s/���2�O-\�\ry��+!�I��u�QZ�4ʨ�3�1B%,`~�F Three important topics are covered: logic, sets, and functions. 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the foundations: logic and proofs pdf

Example: Prove that ( n + 1) 3 ≥ 3 n if n is a positive integer with n ≤ 4 �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! �N`M����(�(N <> �w@ �L&��/^��ַ�_�wF&e?��l�.��X�;~�DBh&t=�^��ѷ���[Y�i�(��8P[�����]VRGx, stream endobj %PDF-1.5 Revisiting the 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 Ն�_�! endobj x���������o �\$��f3�gٻtQƧe�R�M�PJ��R View 1. 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. In math, CS, and other disciplines, informal proofs which are generally shorter, are generally used. Rules of Inference Section 1.6. 3 0 obj <> /v�ԮO���:��wF|�. m�'b����-��[��{��:vT����6i 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. Z� Rules of Inference Section 1.6. endobj �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�� The Foundations: Logic and Proofs Chapter 1, Part III: Proofs. �j���p��T��C��)i\$#*�Tx3�{���R��z=z4���Ϡ������1�� �-��t0ҭ��r��h�D��!+�|����k� ���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� �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! <> s���1G��5�C!ڶ���j� %���� The Foundations: Logic and Proofs Chapter 1, Part III: Proofs. 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. 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�%��� 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. <>/XObject<>/Font<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 720 540] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> Whereas, in calculus, it is continuous functions of a real variable that are important, such functions are of relatively little interest in discrete mathematics. 1 0 obj stream ��ͤ@��Lsy\$�u �M��#�^��f���T��,I\$h.���2�&������՞�~����h1�k*)6�\$���uV��k����l)�\$� � ��,��_hW��4��QI]� 1��HGf � &G?q4)6c 7{H�T �:"��E`c� endobj 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���>} Functions are sometimes called mappings or transformations. 4 0 obj A B C Students Grades D F Kathy Scott Sandeep Patel �~ox��������`έ຅����>�{pupTI������O�s/���2�O-\�\ry��+!�I��u�QZ�4ʨ�3�1B%,`~�F Three important topics are covered: logic, sets, and functions. C\$G�Tr�Ύ�� �K\y鶋�c������ ���'(�a�����4�l�A`�����or ���y�*��s5����' Mathematicians view it as the opposite of \continuous." �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�>^? [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�� Proofs of Mathematical Statements A proof is a valid argument that establishes the truth of a statement. <> 2 0 obj • More than one rule of inference are often used in a step. 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���� <>>> 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���=®����������[� 0 Logic and Proofs.pdf from AA 1THE FOUNDATIONS: LOGIC and PROOFS Foundations of Mathematics Oreste M. Ortega, Jr. Leyte Normal University Foundations of … ;�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� endobj \3D��{��>/��-Z�ϋ��2x�f�)s�h ޓGd�����ߙ� ����+dq`¸�-62�LZPZ1�"�G�PR!IQ�B��\$�.PK�����gm1û4�����O��# J2�&�x�i �u~^O�؅�E�B��i�AO 5 0 obj • Steps may be skipped. 9V%��cD��%�{�ON�� �8�� �,�4��-[���ѽ ���\$�����ݔQu��7aN ٩�� k&�� <>/XObject<>/Font<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 720 540] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> <> 3 0 obj @{�����������Rf6�g�2��=}B���lU <>>> x��V�n�8}7��G�XӼ�*�mn�ER���>}P}UK�d������)�";���\$���9sHQ0|�7o�����߾��������\H�8��h���� ��%� U G�}1x㦠�� ���+f�� � �����\87`f=B��sh��ꣅl���}Zb ꔫ�:E��-�z7ef�YR�#ӹ3�Ԍ-|`ԽVQ�X�� 6 0 obj e�=�*P� endobj endobj 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. endstream %���� 4 0 obj �� ���Pc�MQh| #������U�᭑`���:~\Գ�ÃڲZ �͐ބ}����9дP���>5('f������+( ��}�lW2��n3�w� �^�Q�s������p}��75�hHY��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�� g�����3�o��;��q�p,k��uH!n�5�]���6�|�E�����_I�?����%���m?�f�˧��o%�D��b�?G��'5Km'�. Chapter 1 The Foundations: Logic and Proofs The word \discrete" means separate or distinct. stream Summary Valid Arguments and Rules of Inference Proof Methods Proof Strategies. x��Vێ�6}7��G2�i�IA�dou� 6��Ԇ,�:3s����͛�����۷��� 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��˰� The Foundations: Logic and Proof, Sets, and Functions his chapter reviews the foundations of discrete mathematics. 2 0 obj endobj 1 0 obj %PDF-1.5 <> �l+�M��E���-���"i�����X����P+�,�} N�x��m�a��,��̵�w�F�;",��;��E���X�۶c�H@̈́n��«��"��%���@�|�L+*,N�Ж|H�%�� ���cZ"H\$8B�)Mv���g�`�3�U�D�?�j�ٰČI�F��V.���

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