# multiple comparison t test

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