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