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A B C Students Grades D F Kathy Scott Sandeep Patel s���1G��5�C!ڶ���j� �N`M����(�(N x���������o �$��f3�gٻtQƧe�R�M�PJ��R <> Rules of Inference Section 1.6. 1 0 obj TUQ����y��Mg,��`}��ś���k�ׯ���}�N@_��{G���s찚B�'�T�.3��po\-���4���-��sт@ �[ה���K��{�u��Xq�3��IE#�uԈ���G�� �+Av�Rb � ���o�w o�D`�[�Û4- �� �K�NSj�� ڴSk�ro�������EH�~�E�V"��\ %DzD�v��1�[�CЎ ��T'��1��f�=z0��1�> M�����H�߆�\�s��=x�vI�ȹh~����1p�^݃fS&��Q7G=�>^bʥ,�a�R�f���-����R���t��ҷ�Z���O�i�dæ���� ����+dq`¸�-62�LZPZ1�"�G�PR!IQ�B��$�.PK�����gm1û4�����O��# J2�&�x�i �u~^O�؅�E�B��i�AO ٩�� k&�� 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. <> Three important topics are covered: logic, sets, and functions. ��%� U G�}1x㦠�� ���+f�� � �����\87`f=B��sh��ꣅl���}Zb ꔫ�:E��-�z7ef�YR�#ӹ3�Ԍ-|`ԽVQ�X�� <>/XObject<>/Font<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 720 540] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> ���cZ"H$8B�)Mv���g�`�3�U�D�?�j�ٰČI�F��V.��� <> Mathematicians view it as the opposite of \continuous." �8p�9RrNr0�C����l8�}1�*���s+�n�����O���_4*�W����=���O��ja�:�����^ �Lr|h�C���PD=�)�������u.8�����絥Q�%Q�Lk�I�P��!�� �u��S�� Ն�_�! 4 0 obj View 1. Functions are sometimes called mappings or transformations. Section Summary Valid Arguments Inference Rules for Propositional Logic Using Rules of Inference to Build Arguments Rules of Inference for Quantified Statements Building Arguments for Quantified Statements. �Æ��(�yt_`�;|C9�BxO����VѱBT b ֱ��wnj�u����n�) ���!C��]>�6�ӱE�,D=-�����g_���� ���H�D����/ ��x��nZ��FT"�E�?�x���QO��� 95��ע�f�' �iS��x�2Ơu�x�F�~ ���e�7�ȼ��:Xm�1.є�N4ͱ��޲�Ê�:2�x��QO�܈�������-s���_�V�m�D��# '���7,>�T�>^? endobj 1 The Foundations: Logic and Proofs 1.1 Propositional Logic 1. a proposition is a declarative sentence that is either true (T) or false (F), but not both. �l+�M��E���-���"i�����X����P+�,�} N�x��m�a��,��̵�w�F�;",��;��E���X�۶c�H@̈́n��«��"��%���@�|�L+*,N�Ж|H�%�� Whereas, in calculus, it is continuous functions of a real variable that are important, such functions are of relatively little interest in discrete mathematics. Rules of Inference Section 1.6. %���� 5 0 obj C� Kh8�/�@�T#�s�@���W�m�z�~`9��ܦ6�!�:s�������k���JKߕ�Xj:����p-[���u¸b>L'�wZ�W�x��?Ǒ�� �c��S�����s�rcl"� �0p̚N����0���g�Tg��۷�"��J��˰� <> endobj ��,��_hW��4��QI]� 1��HGf � &G?q4)6c 7{H�T �:"��E`c� ;�eIޖ塕�[�N��8����'�}�/�������@ �@ �@ �Xb�)=M ,19�0k�LI�z��B��U�z��߼j��,@j]�F�l�Cc̈́�����5��Z� �Kb?���ঔ�����Z").v}�\hd�Nł�����M��jDD�vR[$̣{X�Agh���C���T#���9.L� 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. <>>> 2 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� %���� %PDF-1.5 endobj �� ���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�� 1 0 obj ��ͤ@��Lsy$�u �M��#�^��f���T��,I$h.���2�&������՞�~����h1�k*)6�$���uV��k����l)�$� � Revisiting the <> 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�%��� Example: Prove that ( n + 1) 3 ≥ 3 n if n is a positive integer with n ≤ 4 6 0 obj 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 • More than one rule of inference are often used in a step. stream 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.
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