0000013583 00000 n 0000001300 00000 n Dedekind was the rst to introduce SET A. 0000011676 00000 n trailer << /Size 79 /Info 34 0 R /Root 37 0 R /Prev 208577 /ID[<677ee73ff2878ac1943734af0e1128fe>] >> startxref 0 %%EOF 37 0 obj << /Type /Catalog /Pages 33 0 R /Metadata 35 0 R /PageLabels 32 0 R >> endobj 77 0 obj << /S 349 /L 463 /Filter /FlateDecode /Length 78 0 R >> stream The Improving Mathematics Education in Schools (TIMES) Project . H�b```f``a`c`�� Ā B�@Q� P ����S�Lg�RF�e֦�? 0000003978 00000 n ]����]>2-�y%��A�œ.�u������o�T�u>�h�E�4��m������7K��3�W0cY�1O��-�? 0000006005 00000 n 0000008708 00000 n 0000001207 00000 n (Georg Cantor) In the previous chapters, we have often encountered "sets", for example, prime numbers form a set, domains in predicate logic form sets as well. itive concepts of set theory the words “class”, “set” and “belong to”. Dedekind is mostly known for his research in algebra and set theory. 0000121046 00000 n 0000010416 00000 n %PDF-1.3 %���� In this chapter we will outline the background needed for a course in abstract algebra… of abstract algebra. 0000008254 00000 n 0000003690 00000 n De ning a set formally is a pretty delicate matter, for now, we will be happy We then present and briefly dis-cuss the fundamental Zermelo-Fraenkel axioms of set theory. NUMBER AND ALGEBRA . 0000007380 00000 n A Little Set Theory (Never Hurt Anybody) Matthew Saltzman Department of Mathematical Sciences Clemson University Draft: August 19, 2019 1 Introduction The fundamental ideas of set theory and the algebra of sets are probably the most important concepts across all areas of mathematics other than the algebra of real numbers. 1. 0000055893 00000 n 0000008040 00000 n 1.1. 0000001785 00000 n Sets and elements Set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. Ling 310, adapted from UMass Ling 409, Partee lecture notes March 1, 2006 p. 4 Set Theory Basics.doc 1.4. These will be the only primitive concepts in our system. Set Theory \A set is a Many that allows itself to be thought of as a One." H��TMo�0��W�і6���#�nH�� ��C�z�Em�ڤB�{����J���9�؞�7���˶8k[M��U�o�!�ogj/�#ﴨ���}(��'G���$M�P�}�Rt;�RJKm_D�ߋkv��N6�R�a�K�\Pw��݆�s�k�����l�%y��p�k���e7倉.��m�1�IƂ>u�9J�������1�!���m?� [׊J%�U��w��׮m.���F0�zTm�� &� �,�`���D��UF(++|��Ҫ��Y�"R2+����Lm� F�0n�u�D�e�����$n��ˊ� 2�,���ml�qo��^��XˬdDWn߲R��%@�@:�����a3O�jF㊮���b��.��U�yd��Ϙ�9��g}y`,���ZBl^���0��RϨc����>��4�shy7�qB|n8Y�!/Q�6ё����AG�kj�͑Y��p%h�EN�� &r�r=������,�B�tj8�f9CZ4��G��=Tw�/\6��$"�N�7�~��u�35��Е��~�^��J��~~�i �T�H�K:6�r�j/�>*��^Ii�Jr�I���i"�o�t�#�T��(*��R�)�U��3����2_"o�1/��kD�m�K�Nu��j垾M�A5w��c�mB�^�b9��Q����B��q��uӸkQ�u�DZg�������h�S� �ۛ endstream endobj 47 0 obj 488 endobj 48 0 obj << /Filter /FlateDecode /Length 47 0 R >> stream ]Z��. He was the rst to de ne real numbers by means of cuts of rational numbers. To this day many schools around the globe teach the theory of real numbers based on Dedekind’s cuts. 0000003918 00000 n Sets and Venn diagrams (Number and Algebra : Module 1) For teachers of Primary and Secondary Mathematics. 0000005345 00000 n Basic Concepts of Set Theory. 0000009963 00000 n 0000010679 00000 n applications of abstract algebra. 0000012622 00000 n (Caution: sometimes ⊂ is used the way we are using ⊆.) Solution - Pure Mathematics for Beginners: A Rigorous Introduction to Logic, Set Theory, Abstract Algebra, Number Theory, Real Analysis, Topology, Complex Analysis, and Linear Algebra … 510 Cover design, Layout design and Typesetting by Claire Ho. 0000096302 00000 n 1.1 Contradictory statements. Module 1. 0000078392 00000 n ���E�]gZ��I�����5�����A� 0000012643 00000 n June 2011. In this chapter we will outline the background needed for a course in abstract algebra. 0000004758 00000 n A guide for teachers - Years 7–8. 1�t�1��@�DD��%�$�((���Ȓ� 2�q�PZ��Yy�Vb-�Q�]N2�d|�p�L߁X������$���p�� �F��@�]� w��n ��ӹ,g兟s}�0�#�B�!��.c*��J�4{ ��z( endstream endobj 78 0 obj 372 endobj 38 0 obj << /Type /Page /Parent 33 0 R /Resources 39 0 R /Contents [ 46 0 R 48 0 R 50 0 R 52 0 R 60 0 R 64 0 R 66 0 R 68 0 R ] /MediaBox [ 0 0 595 842 ] /CropBox [ 0 0 595 842 ] /Rotate 0 >> endobj 39 0 obj << /ProcSet [ /PDF /Text ] /Font << /TT2 41 0 R /TT4 42 0 R /TT6 53 0 R /TT8 54 0 R /TT10 57 0 R /TT12 61 0 R >> /ExtGState << /GS1 70 0 R >> /ColorSpace << /Cs6 44 0 R >> >> endobj 40 0 obj << /Type /FontDescriptor /Ascent 891 /CapHeight 656 /Descent -216 /Flags 34 /FontBBox [ -558 -307 2000 1026 ] /FontName /FFCPFL+TimesNewRoman,Bold /ItalicAngle 0 /StemV 133 /FontFile2 72 0 R >> endobj 41 0 obj << /Type /Font /Subtype /TrueType /FirstChar 32 /LastChar 233 /Widths [ 250 0 0 0 0 0 0 0 333 333 0 564 250 333 250 278 500 500 500 500 500 500 500 500 500 500 278 278 564 564 564 444 0 722 667 667 722 611 556 722 722 333 0 722 611 889 722 722 556 722 667 556 611 722 0 944 0 722 0 333 0 333 0 0 0 444 500 444 500 444 333 500 500 278 278 500 278 778 500 500 500 500 333 389 278 500 500 722 500 500 444 480 200 480 0 0 0 0 0 0 0 1000 0 0 0 0 0 0 0 0 0 0 0 333 333 444 444 0 500 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 444 ] /Encoding /WinAnsiEncoding /BaseFont /FFCPAJ+TimesNewRoman /FontDescriptor 43 0 R >> endobj 42 0 obj << /Type /Font /Subtype /TrueType /FirstChar 32 /LastChar 148 /Widths [ 250 333 0 0 0 0 0 0 0 0 0 0 250 333 250 0 0 0 0 0 0 0 0 0 0 0 333 0 0 0 0 500 0 0 667 722 722 667 611 0 0 0 0 0 0 0 722 0 0 0 722 556 667 0 0 1000 0 0 0 0 0 0 0 0 0 500 556 444 556 444 333 500 556 278 333 0 278 833 556 500 556 556 444 389 333 556 500 722 0 500 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 500 500 ] /Encoding /WinAnsiEncoding /BaseFont /FFCPFL+TimesNewRoman,Bold /FontDescriptor 40 0 R >> endobj 43 0 obj << /Type /FontDescriptor /Ascent 891 /CapHeight 656 /Descent -216 /Flags 34 /FontBBox [ -568 -307 2000 1007 ] /FontName /FFCPAJ+TimesNewRoman /ItalicAngle 0 /StemV 94 /XHeight 0 /FontFile2 71 0 R >> endobj 44 0 obj [ /ICCBased 69 0 R ] endobj 45 0 obj 702 endobj 46 0 obj << /Filter /FlateDecode /Length 45 0 R >> stream A basic knowledge of set theory, mathe-matical induction, equivalence relations, and matrices is a must. 0000002189 00000 n 0000003957 00000 n When expressed in a mathematical context, the word “statement” is viewed in a 0000006827 00000 n Subsets A set A is a subset of a set B iff every element of A is also an element of B.Such a relation between sets is denoted by A ⊆ B.If A ⊆ B and A ≠ B we call A a proper subset of B and write A ⊂ B. 0000139607 00000 n 1.1 A Short Note on Proofs 0000008932 00000 n 0000011655 00000 n 0000001764 00000 n Even more important is the ability to read and understand mathematical proofs. 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