endstream << x���P(�� �� /Resources 10 0 R >> In this module we’ve seen how logic and valid arguments can be formalized using mathematical notation and a few basic rules. endobj /Type /XObject /Subtype /Form �y�,��{�̒����_��-г�������� << Informal Proof. 156 0 obj /Type /XObject /Subtype /Form 9 0 obj George Boole. x��Y�n�F��+�� �y?���6���C�e�|�$UG�;��dE$�-wcR#Υ�s�=3��Ie0� e�d$P��B�`:��YU��l����ݫs��8e&B@����0���e����. /Resources 21 0 R Predicate Logic and Quantifiers. %���� /Matrix [1 0 0 1 0 0] 11 0 obj /BBox [0 0 100 100] /Matrix [1 0 0 1 0 0] x���P(�� �� /Filter /FlateDecode That is, we admit, as a starting point, the existence of certain objects (which we call sets), which we won’t define, but which we assume satisfy some basic properties, which we express as axioms. /Resources 24 0 R x���P(�� �� /FormType 1 stream ����BK1yӓ� ��κv�������*�u��cvIr��mxzz(���2~ˤ��͉r$�����%T=� �e�*P4�R����� ^C)�w�S ��5����ùo�ӔRP���Y�R��q=�����)}�T�릧6��)VH��xG�~tM�-�I/$��y���=�ό����Aö��'Cw��N�E�̈��E����@-��1BS�F{��{T+r��9js�f�O�k�O�������=�k1�U9���A��Wpԍ(��1-�HK�Q�+K����u��"n�}�+Ȍ��2�~��s�x�+]���oaCT��{#�Y�õP�C�F��W��[�km endobj stream >> Complex issues arise in Set Theory more than any other area of pure mathematics; in particular, Mathematical Logic is used in a fundamental way. << Module 6: Set Theory and Logic. /Subtype /Form endstream >> endstream >> is the formal systematic study ofthe principles of valid inference andcorrect reasoning. Prepared by:Nathaniel T. SullanoBS Math – 3 2. is a science that deals with the principlesand criteria of validity of inference anddemonstration. /Filter /FlateDecode /Length 15 x���P(�� �� /Length 15 endstream /Length 15 logic and set theory 1. No speci c prerequisites. stream The mathematical logic group offers thesis projects in all mathematical study programmes at Bonn. 7 0 obj /Filter /FlateDecode IV. /Matrix [1 0 0 1 0 0] >> It has been and is likely to continue to be a a source of fundamental ideas in Computer Science from theory to practice; Computer Science, being a science of the arti cial, has had many of its constructs and ideas inspired by Set Theory. ۖȄnKt|ѭ��8��~�ɩ�1ƒ�v���C�v�(*Ɇ����"С� �����5�|w�D�Ķi�~�kSG%|��~b3��K��1���E���h�s�|�Hq3�@�h���L�����ZQs\��J3�f�A�Rd��qݼК[ג�t��4ˌ���M2�@أ����i��ƚ�I0����M��W��j�*~���ْ+ �Wz�5��pM���H��]/Îc�S+�@?��_sW ����-+�2�RJ/:�&���O>l~�R�}J}�U�O�}�jC�v�b/�X�8�'�LD�^�)���Ǵv��������#�Ύ|�A����7 0+��(�D��m Logic and Set Theory. /BBox [0 0 100 100] /Type /XObject /Length 1109 The information pertaining to the courses on Logic and Set Theory (2IT60 and 2IHT10) that I teach is now (only) available from the respective Canvas sites. /Filter /FlateDecode /Filter /FlateDecode /Matrix [1 0 0 1 0 0] /FormType 1 /Length 15 Indirect Proof. x���P(�� �� /BBox [0 0 100 100] /FormType 1 /Subtype /Form x���P(�� �� x���P(�� �� In mathematics, the notion of a set is a primitive notion. stream /Filter /FlateDecode Predicates. �a6w�Y�U�tHLh_����4؉���X�&�2@7����H�yvg���m{ stream Please go to canvas.tue.nl. Studying mathematical logic and set theory at Bonn. xڍW�r�6}�W�T�B ��yjҴu'I3u:'0I�Tl���{ʲb7}`{9{�L�m�D�>K�1�\=���F�J�,Zm"Yj�:��L�2�V��2~�Pyl�i�L_,�����xx�o]�Sӭy��z�v���W��,�RTZ+�Mi��4Z�R����^�ђ��w4qc��0ؑW�X��������o�Yہ�����������e������oz�O^g���\/Tێ?�胮���g{;A�_����B��1wR%t^��T�2��:a��v�Dj��*PUD��H$iZJ���T�L�(Kɺ��B���.؎���h��ƺ8�X�Z��w�ig�H�0~�t��;�PU��D����l�k�)�٬�Z�}Jt�o&x��DTeǒHF2C��*�:�UWF�����tt���A�$]el6qL��#�YQR������$���k�����`��)l v]�� /BBox [0 0 100 100] 17 0 obj Leader Notes taken by Dexter Chua Lent 2015 These notes are not endorsed by the lecturers, and I have modi ed them (often signi cantly) after lectures. 20 0 obj << Formal Proof. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine. 4 0 obj << %PDF-1.5 >> V. Naïve Set Theory. /Type /XObject stream /Type /XObject Set Theory and Logic: Fundamental Concepts (Notes by Dr. J. Santos) A.1. B. Universal and Existential Quantifiers. endobj << /Type /XObject /Subtype /Form Set Theory is indivisible from Logic where Computer Science has its roots. /BBox [0 0 100 100] /Matrix [1 0 0 1 0 0] x���P(�� �� /Filter /FlateDecode /FormType 1 << Proof by Counter Example. << Multiple Quantifiers. /FormType 1 >> Like logic, the subject of sets is rich and interesting for its own sake. /Length 15 /Length 15 /Length 15 /Resources 5 0 R stream 23 0 obj /Filter /FlateDecode /Resources 8 0 R Besides the general rules and regulations for these programs, the following pages provide some logic-specific information. /Subtype /Form Although Elementary Set Theory is well-known and straightforward, the modern subject, Axiomatic Set Theory, is both conceptually more difficult and more interesting. Negation of Quantified Predicates. >> ~ �^�O�^�צ�cr�~+���5�Pߚ�3Hr�Vw���� +�i�4�9^�����=G�E�*ı ��8��+ �4�?����{6t���[�+� �y~Q�1T�:+�����"��5]h��y�����N�t�O�e�9��p �����6�ػ���KY�(�a*�a��������sT�P�.��D�U~�zNux�>L0� =� 200 0 obj endobj << >> Part II | Logic and Set Theory Based on lectures by I. /Filter /FlateDecode /FormType 1 endobj endstream )� �XʸǬ�H���fg&���#�:+D�2v�4����j`�i��`������v�ȳ4���lj�#�Ϸ����aY%'�_��Hs�~)�T�؇P^��~�R�Ӹ�|�z�ZB�ݿnt%�oHOnH��Я�u`�ǰv�i��j���qm���KT�DͱV�HSe ����(��Y�j�u[R�� �c����ӓ�jl� endstream endobj endobj 26 0 obj endobj Primitive Concepts. >> /Type /XObject Mathematical Induction. We will need only a few facts about sets and techniques for dealing with them, which we set out in this section and the next. III. /FormType 1 endstream /BBox [0 0 100 100] stream endstream /Resources 12 0 R /Resources 18 0 R /Subtype /Form /FormType 1 stream Students from other fields may take logic courses within their secondary subjects or as optional modules. /Type /XObject endstream endobj /Length 1777 /Matrix [1 0 0 1 0 0] /BBox [0 0 100 100] Unique Existence. /Matrix [1 0 0 1 0 0] /BBox [0 0 100 100] << /Filter /FlateDecode Conditional Proof. /Length 15 Methods of Proof. /Resources 27 0 R /Subtype /Form stream /Matrix [1 0 0 1 0 0] Search for: Putting It Together: Set Theory and Logic.

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